Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.110:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Question 14:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Question 18:
Answer:
Page No 19.111:
Question 19:
Answer:
Page No 19.111:
Question 20:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Answer:
Page No 19.111:
Question 25:
Answer:
Page No 19.111:
Question 26:
Answer:
Page No 19.112:
Answer:
Page No 19.112:
Answer:
Page No 19.112:
Answer:
Page No 19.112:
Answer:
Page No 19.112:
Answer:
Page No 19.112:
Question 32:
Answer:
Page No 19.112:
Question 33:
Evaluate the following integrals as limit of sums:
[CBSE 2014]
Answer:
We have,
Here,
a = 1,
b = 3,
f(
x) = 3
x2 + 1 and
Page No 19.112:
Question 34:
Answer:
Page No 19.115:
Answer:
Page No 19.115:
Answer:
Page No 19.115:
Answer:
Page No 19.115:
Question 4:
Answer:
Page No 19.115:
Question 5:
Answer:
Page No 19.115:
Question 6:
Answer:
Page No 19.115:
Question 7:
Answer:
Page No 19.115:
Question 8:
Answer:
Page No 19.116:
Answer:
Page No 19.116:
Question 10:
Answer:
Page No 19.116:
Question 11:
Answer:
Page No 19.116:
Question 12:
Answer:
Page No 19.116:
Question 13:
Answer:
Page No 19.116:
Question 14:
Answer:
Page No 19.116:
Question 15:
Answer:
I =
using partial fraction,
putting the values of A,B and C we get
Page No 19.116:
Question 16:
Answer:
Page No 19.116:
Question 17:
Answer:
Page No 19.116:
Question 18:
Answer:
Page No 19.116:
Question 19:
Answer:
Page No 19.116:
Question 20:
Answer:
Page No 19.116:
Question 21:
Answer:
Page No 19.116:
Question 22:
Answer:
Page No 19.116:
Question 23:
Evaluate the following integrals:
Answer:
Let
I =
Put 2
x + 1 =
z2
When
When
Page No 19.116:
Question 24:
Answer:
Page No 19.116:
Question 25:
Answer:
Page No 19.116:
Question 26:
Answer:
Page No 19.116:
Question 27:
Answer:
Page No 19.116:
Question 28:
Answer:
Page No 19.116:
Answer:
Page No 19.116:
Answer:
Page No 19.116:
Question 31:
Answer:
Page No 19.116:
Question 32:
Answer:
Page No 19.116:
Answer:
Page No 19.116:
Question 34:
Answer:
Page No 19.116:
Question 35:
Answer:
Page No 19.116:
Question 36:
Answer:
Page No 19.116:
Question 37:
Answer:
Page No 19.116:
Question 38:
Answer:
Page No 19.116:
Question 39:
Answer:
Page No 19.116:
Question 40:
Answer:
Page No 19.116:
Question 41:
Answer:
Page No 19.117:
Question 42:
Answer:
Page No 19.117:
Question 43:
Answer:
Page No 19.117:
Question 44:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Question 46:
Answer:
Page No 19.117:
Question 47:
Answer:
Page No 19.117:
Question 48:
Answer:
Page No 19.117:
Question 49:
Answer:
Page No 19.117:
Question 50:
Answer:
Page No 19.117:
Question 51:
Answer:
Page No 19.117:
Question 52:
Answer:
Page No 19.117:
Question 53:
Answer:
Page No 19.117:
Question 54:
Answer:
Page No 19.117:
Question 55:
Answer:
Page No 19.117:
Question 56:
Answer:
Page No 19.117:
Question 57:
Answer:
Page No 19.117:
Question 58:
Answer:
Page No 19.117:
Question 59:
Answer:
Page No 19.117:
Question 60:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Question 66:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.117:
Answer:
Page No 19.118:
Question 1:
equals
(a) π/2
(b) π/4
(c) π/6
(d) π/8
Answer:
(d) /8
Page No 19.118:
Question 2:
equals
(a) 0
(b) 1/2
(c) 2
(d) 3/2
Answer:
(c) 2
Page No 19.118:
Question 3:
The value of is
(a)
(b)
(c)
(d)
Answer:
π24
Page No 19.119:
Question 4:
The value of is
(a) 0
(b) 2
(c) 8
(d) 4
Answer:
(c) 8
Page No 19.119:
Question 5:
The value of the integral is
(a) 0
(b) π/2
(c) π/4
(d) none of these
Answer:
(c) π/4
Page No 19.119:
Question 6:
equals
(a) log 2 − 1
(b) log 2
(c) log 4 − 1
(d) − log 2
Answer:
(b) log 2
Page No 19.119:
Question 7:
equals
(a) 2
(b) 1
(c) π/4
(d) π
2/8
Answer:
(a) 2
Page No 19.119:
Question 8:
equals
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 19.119:
Question 9:
equals
(a)
(b)
(c)
(d)
Answer:
3√tan−1(13√)
Page No 19.119:
Question 10:
(a)
(b)
(c)
(d) π + 1
Answer:
Disclaimer: None of the given option is correct.
Page No 19.119:
Question 11:
(a)
(b)
(c)
(d) (
a +
b) π
Answer:
Page No 19.119:
Question 12:
is
(a) π/3
(b) π/6
(c) π/12
(d) π/2
Answer:
Page No 19.119:
Question 13:
Given that the value of is
(a)
(b)
(c)
(d)
Answer:
Page No 19.120:
Question 14:
(a) 1
(b)
e − 1
(c)
e + 1
(d) 0
Answer:
(a) 1
Page No 19.120:
Question 15:
is equal to
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 19.120:
Question 16:
(a)
(b)
(c)
(d)
Answer:
Page No 19.120:
Question 17:
The value of the integral is
(a)
(b)
(c)
(d)
Answer:
Page No 19.120:
Question 18:
is equal to
(a) 1
(b) 2
(c) − 1
(d) − 2
Answer:
(b) 2
Page No 19.120:
Question 19:
is equal to
(a)
(b)
(c)
(d) π
Answer:
(a)
Page No 19.120:
Question 20:
The value of is
(a) 1
(b) e − 1
(c) 0
(d) − 1
Answer:
(b) e − 1
Page No 19.120:
Question 21:
If then a equals
(a)
(b)
(c)
(d) 1
Answer:
(b)
Page No 19.120:
Question 22:
If equals
(a) 4a2
(b) 0
(c) 2a2
(d) none of these
Answer:
(b) 0
Page No 19.120:
Question 23:
The value of is
(a)
(b)
(c) 0
(d) none of these
Answer:
(c) 0
Page No 19.121:
Question 24:
is equal to
(a) log
e 3
(b)
(c)
(d) log (−1)
Answer:
(b)
Page No 19.121:
Question 25:
is equal to
(a) −2
(b) 2
(c) 0
(d) 4
Answer:
(b) 2
Page No 19.121:
Question 26:
The derivative of is
(a)
(b)
(c) (ln x)−1x (x − 1)
(d)
Answer:
(c) (ln x)−1x (x − 1)
Using Newton Leibnitz formula
Page No 19.121:
Question 27:
If then the value of I10 + 90I8 is
(a)
(b)
(c)
(d)
Answer:
Page No 19.121:
Question 28:
(a)
(b)
(c)
(d)
Answer:
Disclaimer: The question given is not correct because the function provided does not converge in the given domain.
Page No 19.121:
Question 29:
is equal to
(a)
(b)
(c) 2
(b)
Answer:
Hence, the correct option is (d).
Page No 19.121:
Question 30:
The value of the integral is
(a) 4
(b) 2
(c) −2
(d) 0
Answer:
(a) 4
Page No 19.121:
Question 31:
is equal to
(a) 0
(b) 1
(c) π/2
(d) π/4
Answer:
(d) π/4
Page No 19.121:
Question 32:
equals to
(a) π
(b) π/2
(c) π/3
(d) π/4
Answer:
(d) π/4
Page No 19.121:
Question 33:
is equal to
(a) 0
(b) π
(c) π/2
(d) π/4
Answer:
(c) π/2
Page No 19.122:
Question 34:
is equal to
(a) π/4
(b) π/2
(c) π
(d) 1
Answer:
(d) 1
Page No 19.122:
Question 35:
is equal to
(a) π
(b) π/2
(c) 0
(d) 2π
Answer:
(c) 0
Page No 19.122:
Question 36:
The value of is
(a) π/4
(b) π/8
(c) π/2
(d) 0
Answer:
(a) π/4
Page No 19.122:
Question 37:
(a) π ln 2
(b) −π ln 2
(c) 0
(d)
Answer:
(a) π ln 2
Substitute x = tan θ
⇒ dx = sec2 θ dθ.
when,
x = 0 ⇒ θ = 0
Let us consider,
Page No 19.122:
Question 38:
is equal to
(a)
(b) 0
(c)
(d)
Answer:
Page No 19.122:
Question 39:
If f (a + b − x) = f (x), then x f (x) dx is equal to
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 19.122:
Question 40:
The value of is
(a) 1
(b) 0
(c) −1
(d) π/4
Answer:
(b) 0
Page No 19.122:
Question 41:
The value of is
(a) 2
(b)
(c) 0
(d) −2
Answer:
(c) 0
Page No 19.122:
Question 42:
The value of is
(a) 0
(b) 2
(c) π
(d) 1
Answer:
(c)
Page No 19.123:
Question 43:
is equal to
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
â
Hence, the correct option is (a).
Page No 19.123:
Question 44:
is equal to
(a)
(b)
(c)
(d)
Answer:
â
Hence, the correct option is (b).
Page No 19.123:
Question 45:
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) = g(a – x) = a, then is equal to
(a)
(b)
(c)
(d)
Answer:
Given: f(x) = f(a – x) and g(x) + g(a – x) = a
â
Hence, the correct option is (b).
Page No 19.123:
Question 1:
If then a = _______________.
Answer:
â
Hence,
a =
.
Page No 19.123:
Question 2:
The value of is _______________.
Answer:
â
Hence, the value of
is
0.
Page No 19.123:
Question 3:
The value of is _______________.
Answer:
â
Hence, the value of
Page No 19.123:
Question 4:
_______________.
Answer:
â
Hence,
Page No 19.123:
Question 5:
The value of the integral is _______________.
Answer:
â
Hence, the value of the integral
is
1.
Page No 19.123:
Question 6:
The value of the integral is _______________.
Answer:
â
Hence, the value of the integral
Page No 19.123:
Question 7:
The value of the integral is _______________.
Answer:
â
Hence, the value of the integral
is
Page No 19.124:
Question 8:
________________.
Answer:
â
Hence,
.
Page No 19.124:
Question 9:
The value of the integral where a, b, c, d are constants, depends only on ________________.
Answer:
âHence, the value of the integral
where
a,
b,
c,
d are constants, depends only on
d.
Page No 19.124:
Question 10:
________________.
Answer:
âHence,
2.
Page No 19.124:
Question 11:
________________.
Answer:
âHence,
Page No 19.124:
Question 12:
________________.
Answer:
âHence,
Page No 19.124:
Question 13:
The value of is ________________.
Answer:
âHence, the value of
is
Page No 19.124:
Question 14:
The value of is ________________.
Answer:
âHence, the value of
is
.
Page No 19.124:
Question 15:
The value of is ________________.
Answer:
âHence, the value of
is
.
Page No 19.124:
Question 16:
If f(a – x) = x and then k = _____________.
Answer:
Given:
f(
x) =
f(
a –
x) ...(1)
...(2)
â
âHence,
k =
2.
Page No 19.124:
Question 17:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
0.
Page No 19.124:
Question 18:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
0.
Page No 19.124:
Question 19:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
0.
Page No 19.124:
Question 20:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
2.
Page No 19.124:
Question 21:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
.
Page No 19.124:
Question 22:
If then k = ________________.
Answer:
â
âHence,
k =
a.
Page No 19.124:
Question 23:
If f(x) = f(a – x) and then k = ________________.
Answer:
Given:
f(x) = f(a – x) ...(1)
...(2)
â
âHence, k = .
Page No 19.125:
Question 24:
The value of the integral is ________________.
Answer:
â
âHence, the value of the integral
is
5.
Page No 19.125:
Question 25:
________________.
Answer:
â
âHence,
Page No 19.125:
Question 1:
Answer:
Page No 19.125:
Question 2:
Answer:
Page No 19.125:
Question 3:
Answer:
Page No 19.125:
Question 4:
Answer:
Page No 19.125:
Question 5:
Answer:
Page No 19.125:
Question 6:
Answer:
Page No 19.125:
Question 7:
Answer:
Page No 19.125:
Answer:
Page No 19.125:
Answer:
Page No 19.125:
Answer:
Page No 19.125:
Question 11:
Answer:
Page No 19.125:
Question 12:
Answer:
Page No 19.125:
Question 13:
Answer:
Page No 19.125:
Question 14:
Answer:
Page No 19.125:
Question 15:
Answer:
Page No 19.125:
Question 16:
Answer:
Page No 19.126:
Question 17:
Answer:
Page No 19.126:
Question 18:
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Question 20:
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Question 22:
Evaluate each of the following integrals:
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Question 25:
Answer:
Page No 19.126:
Question 26:
Evaluate each of the following integrals:
[CBSE 2014]
Answer:
Put
When
When
Page No 19.126:
Question 27:
Evaluate each of the following integrals:
[CBSE 2014]
Answer:
Page No 19.126:
Question 28:
Evaluate each of the following integrals:
[CBSE 2014]
Answer:
Page No 19.126:
Question 29:
Evaluate each of the following integrals:
[CBSE 2014]
Answer:
Disclaimer: The solution has been provided by taking the lower limit of integral as 0.
Page No 19.126:
Question 30:
Solve each of the following integrals:
[CBSE 2014]
Answer:
Page No 19.126:
Question 31:
If find the value of k.
Answer:
Page No 19.126:
Question 32:
If write the value of a.
Answer:
Page No 19.126:
Question 33:
If , the write the value of . [CBSE 2014]
Answer:
Differentiating both sides with respect to
x, we get
Thus, the value of
is
x sin
x.
Page No 19.126:
Question 34:
If , find the value of a. [CBSE 2014]
Answer:
Thus, the value of
a is 2.
Page No 19.126:
Question 35:
Write the coefficient a, b, c of which the value of the integral is independent.
Answer:
Hence, the given integral is independent of
b
Page No 19.126:
Question 36:
Evaluate :
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Answer:
Page No 19.126:
Question 39:
where {
x} denotes the fractional part of
x.
Answer:
Page No 19.126:
Answer:
Page No 19.127:
Answer:
Page No 19.127:
Answer:
Page No 19.127:
Question 43:
Answer:
Page No 19.127:
Answer:
Page No 19.127:
Question 45:
If denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Question 3:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Question 6:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Answer:
Page No 19.16:
Question 10:
Answer:
Page No 19.16:
Question 11:
Answer:
Page No 19.16:
Question 12:
Answer:
Page No 19.16:
Question 13:
Answer:
Page No 19.16:
Question 14:
Answer:
Page No 19.16:
Question 15:
Answer:
Page No 19.16:
Question 16:
Answer:
Page No 19.16:
Question 17:
Answer:
Page No 19.16:
Question 18:
Answer:
Page No 19.16:
Question 19:
Answer:
Page No 19.16:
Question 20:
Answer:
Page No 19.16:
Question 21:
Answer:
Page No 19.16:
Question 22:
Answer:
Page No 19.16:
Question 23:
Answer:
Page No 19.16:
Question 24:
Answer:
Page No 19.16:
Question 25:
Answer:
Page No 19.16:
Question 26:
Evaluate the following definite integrals:
[CBSE 2014]
Answer:
Applying integration by parts, we have
Again applying integration by parts, we have
Page No 19.17:
Question 27:
Answer:
Page No 19.17:
Question 28:
Answer:
Page No 19.17:
Question 29:
Answer:
Page No 19.17:
Question 30:
Answer:
Page No 19.17:
Question 31:
Answer:
Page No 19.17:
Answer:
Page No 19.17:
Question 33:
Answer:
Page No 19.17:
Question 34:
Answer:
Page No 19.17:
Question 35:
Answer:
Page No 19.17:
Question 36:
Answer:
Page No 19.17:
Question 37:
Answer:
Page No 19.17:
Question 38:
Answer:
Page No 19.17:
Question 39:
Answer:
Page No 19.17:
Question 40:
Answer:
Page No 19.17:
Answer:
Page No 19.17:
Question 42:
Answer:
Page No 19.17:
Question 43:
Answer:
Page No 19.17:
Question 44:
Answer:
Page No 19.17:
Question 45:
Answer:
Page No 19.17:
Question 46:
Answer:
Page No 19.17:
Question 47:
Answer:
Page No 19.17:
Question 48:
Answer:
Page No 19.17:
Question 49:
Answer:
Disclaimer: The answer given in the book has some error. The solution here is created according to the question given in the book.
Page No 19.17:
Question 50:
Answer:
Page No 19.17:
Question 51:
Answer:
Page No 19.17:
Question 52:
Answer:
Page No 19.17:
Question 53:
Evaluate
Answer:
Page No 19.17:
Question 54:
Answer:
Page No 19.17:
Question 55:
Answer:
Page No 19.17:
Question 56:
Answer:
Page No 19.17:
Question 57:
Answer:
Page No 19.17:
Question 58:
Answer:
Page No 19.17:
Question 59:
Evaluate the following definite integrals:
[NCERT EXEMPLAR]
Answer:
Let
I =
Put
Also,
When
When
∴
I =
Page No 19.18:
Question 60:
If find the value of k.
Answer:
Page No 19.18:
Question 61:
If find the value of a.
Answer:
Page No 19.18:
Question 62:
Answer:
Page No 19.18:
Question 63:
Answer:
When
,
Page No 19.18:
Question 64:
Answer:
Page No 19.18:
Question 65:
Answer:
Let
I =
Applying integration by parts, we have
Page No 19.18:
Question 66:
Answer:
Page No 19.18:
Question 67:
Answer:
Page No 19.18:
Question 68:
Answer:
Let
Putting
x = −1, we have
1 = 2
B .....(1)
Putting
x = 0, we have
A +
B +
D = 1 .....(2)
Equating coefficient of
x3 on both sides, we have
A +
C = 0 .....(3)
Equating coefficient of
x2 on both sides, we have
A +
B + 2
C +
D = 0 .....(4)
⇒
2
C = −1 [Using (1)]
[Using (3)]
Putting
and
in (4), we have
D = 0
Page No 19.38:
Answer:
Page No 19.38:
Question 2:
Answer:
Page No 19.38:
Question 3:
Answer:
Page No 19.38:
Question 4:
Answer:
Page No 19.38:
Answer:
Page No 19.38:
Question 6:
Answer:
Page No 19.38:
Answer:
Page No 19.38:
Question 8:
Answer:
Page No 19.38:
Answer:
Page No 19.38:
Answer:
Page No 19.39:
Question 11:
Answer:
Page No 19.39:
Question 12:
Answer:
Page No 19.39:
Question 13:
Answer:
Page No 19.39:
Question 14:
Answer:
Page No 19.39:
Question 15:
Answer:
Page No 19.39:
Answer:
Page No 19.39:
Question 17:
Answer:
Page No 19.39:
Question 18:
Answer:
Page No 19.39:
Question 19:
Answer:
Page No 19.39:
Question 20:
Answer:
Page No 19.39:
Question 21:
Answer:
Page No 19.39:
Question 22:
Answer:
Page No 19.39:
Question 23:
Answer:
Page No 19.39:
Question 24:
Answer:
Let
I =
Put
When
When
Applying integration by parts, we have
Page No 19.39:
Question 25:
Answer:
Page No 19.39:
Question 26:
Answer:
Page No 19.39:
Question 27:
Answer:
Page No 19.39:
Question 28:
Answer:
Page No 19.39:
Question 29:
Answer:
Page No 19.39:
Question 30:
Answer:
Page No 19.39:
Question 31:
Answer:
Let
Put
When
When
Page No 19.39:
Question 32:
Answer:
Page No 19.39:
Question 33:
Answer:
Page No 19.39:
Question 34:
Answer:
Page No 19.39:
Question 35:
Answer:
Page No 19.39:
Question 36:
Answer:
Page No 19.39:
Question 37:
Answer:
Page No 19.39:
Question 38:
Answer:
Page No 19.39:
Question 39:
Answer:
Page No 19.39:
Question 40:
[CBSE 2015]
Answer:
Put tan
x =
z
When
When
Page No 19.39:
Question 41:
Answer:
Page No 19.39:
Question 42:
Answer:
Page No 19.39:
Question 43:
Answer:
Page No 19.39:
Question 44:
Answer:
Page No 19.40:
Question 45:
Answer:
Page No 19.40:
Question 46:
Answer:
Page No 19.40:
Question 47:
Answer:
Page No 19.40:
Question 48:
Answer:
Page No 19.40:
Question 49:
Answer:
Page No 19.40:
Question 50:
Answer:
Page No 19.40:
Question 51:
Answer:
Page No 19.40:
Question 52:
Answer:
Page No 19.40:
Question 53:
Answer:
Page No 19.40:
Question 54:
Answer:
Page No 19.40:
Question 55:
Answer:
Page No 19.40:
Question 56:
Answer:
Page No 19.40:
Question 57:
Answer:
Let
I =
Put
When
When
Page No 19.40:
Question 58:
Answer:
Let
I =
Put
When
When
Dividing numerator and denominator by
, we have
Now, put
When
When
Page No 19.40:
Question 59:
Answer:
Let
I =
Put
When
When
Page No 19.40:
Question 60:
Answer:
Let
Put
When
When
Page No 19.40:
Question 61:
Answer:
Let
I =
Put cos
x =
z2
When
When
Page No 19.40:
Question 62:
Answer:
Let
I =
Put
When
When
Page No 19.55:
Question 1:
(i)
(ii)
(iii)
(iv)
Answer:
(iv)
Hence, = 1.
Page No 19.56:
Question 2:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 3:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 4:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 5:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 6:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 7:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 8:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 9:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 10:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 11:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 12:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 13:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 14:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 15:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 16:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 17:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 18:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 19:
Evaluate the following integrals:
Answer:
Page No 19.56:
Question 20:
Answer:
We know that
When
When
When
Page No 19.56:
Answer:
Consider
.
Now,
⇒
f(
x) is an odd function.
Page No 19.56:
Question 22:
Answer:
Page No 19.56:
Question 23:
Answer:
Consider
Now,
Page No 19.56:
Question 24:
Answer:
Consider
Now,
Page No 19.56:
Question 25:
Answer:
Page No 19.56:
Question 26:
Answer:
Put cos
x =
z2
When
When
Now,
Putting
z = 1, we get
Putting
z = −1, we get
Putting
z = 0, we get
Equating coefficient of
z3 on both sides, we get
Disclaimer: The answer does not matches with the answer provided for the question.
Page No 19.56:
Answer:
Page No 19.56:
Question 28:
Answer:
Page No 19.61:
Question 1:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 2:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 3:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 4:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 5:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Disclaimer: This answer does not matches with the given answer in the book.
Page No 19.61:
Question 6:
Evaluate each of the following integrals:
, a > 0
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 7:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 8:
Evaluate each of the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 9:
Evaluate each of the following integrals:
Answer:
Now,
Consider
.
Let
.
Page No 19.61:
Question 10:
Evaluate each of the following integrals:
Answer:
Let
I =
Then,
Adding (1) and (2), we get
Page No 19.61:
Question 11:
Answer:
Page No 19.61:
Question 12:
Answer:
Page No 19.61:
Question 13:
Answer:
Page No 19.61:
Question 14:
Answer:
Page No 19.61:
Question 15:
Answer:
Page No 19.61:
Question 16:
If , then prove that .
Answer:
Page No 19.95:
Question 1:
Answer:
Page No 19.95:
Question 2:
Answer:
Page No 19.95:
Question 3:
Answer:
â
Page No 19.95:
Question 4:
Answer:
Page No 19.95:
Question 5:
Answer:
Page No 19.95:
Question 6:
Answer:
Page No 19.95:
Question 7:
Answer:
Page No 19.95:
Question 8:
Answer:
Page No 19.95:
Question 9:
Answer:
Page No 19.95:
Question 10:
Answer:
Page No 19.95:
Question 11:
Answer:
Page No 19.95:
Question 12:
Answer:
Page No 19.95:
Question 13:
Answer:
Page No 19.95:
Question 14:
Answer:
Page No 19.95:
Question 15:
Answer:
Page No 19.95:
Question 16:
Answer:
Page No 19.95:
Question 17:
Answer:
Page No 19.95:
Question 18:
Evaluate the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.95:
Question 19:
Evaluate the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Page No 19.95:
Question 20:
Evaluate the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we have
Page No 19.95:
Question 21:
Evaluate the following integrals:
[NCERT EXEMPLAR]
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we have
Page No 19.95:
Question 22:
Answer:
Page No 19.96:
Question 23:
Answer:
Page No 19.96:
Question 24:
Answer:
Page No 19.96:
Question 25:
(i)
(ii)
Answer:
(ii)
Hence, = 0.
Page No 19.96:
Question 26:
Answer:
Page No 19.96:
Question 27:
Answer:
Page No 19.96:
Question 28:
Answer:
Page No 19.96:
Question 29:
Evaluate the following integrals:
Answer:
Let
I =
Then,
Consider
.
Now,
Again, consider
.
Then,
Adding (1) and (2), we get
Put cos
x =
z
When
When
Page No 19.96:
Question 30:
Evaluate the following integrals:
Answer:
Let
I =
Consider
.
Page No 19.96:
Question 31:
Evaluate the following integrals:
Answer:
Let
I =
Consider
.
Now, consider
.
Page No 19.96:
Question 32:
Evaluate the following integrals:
Answer:
Let
I =
Put
When
When
Page No 19.96:
Answer:
Page No 19.96:
Question 34:
Answer:
Page No 19.96:
Question 35:
Evaluate the following integrals:
[NCERT EXEMPLAR]
Answer:
Let
I =
Consider
.
Now,
Page No 19.96:
Question 36:
Evaluate the following integrals:
Answer:
Let
I =
.....(1)
Then,
Adding (1) and (2), we get
Dividing the numerator and denominator by cos
2x, we get
Put tan
x =
z
Then
When
When
Page No 19.96:
Question 37:
Evaluate :
Answer:
Page No 19.96:
Question 38:
Evaluate the following integrals:
Answer:
Let I =
Suppose .
Now,
Again,
Page No 19.96:
Question 39:
Evaluate the following integrals:
Answer:
Let
I =
.....(1)
Then,
.....(2)
Adding (1) and (2), we get
Page No 19.96:
Question 40:
Page No 19.96:
Question 41:
Answer:
Applying the limits, we get
Page No 19.96:
Question 42:
Evaluate :
Answer:
Applying the limits, we get
Page No 19.96:
Question 43:
If f is an integrable function such that f(2a − x) = f(x), then prove that
Answer:
Hence Proved
Page No 19.96:
Question 44:
If f(2a − x) = −f(x), prove that
Answer:
Page No 19.96:
Question 45:
If f is an integrable function, show that
(i)
(ii)
Answer:
(i)
(ii)
Page No 19.96:
Question 46:
If f (x) is a continuous function defined on [0, 2a]. Then, prove that
Answer:
Page No 19.96:
Question 47:
If , then prove that
Answer:
Page No 19.97:
Question 48:
If f(x) is a continuous function defined on [−a, a], then prove that
Answer:
Page No 19.97:
Question 49:
Prove that:
Answer:
Page No 19.97:
Question 50:
Prove that: and hence evaluate
Answer:
To Prove:
Now,
Hence, â
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