Q The value of integral ∫π/43π/4x1+sin x dx is : (a) π2(2+1) (b) π2 - Maths - Integrals

Question

Q The value of integral ∫ π / 4 3 π / 4 x 1 + sin
  x   d x   i s   :        
( a )   π 2 ( 2 + 1 )     ( b )   π 2
    ( c )   2 π ( 2 - 1 )      
( d )   π ( 2 - 1 )

Neha Sethi · Accepted Answer

Dear student
∫π43π4x1+sinxdxConsider, ∫x1+sinxdx=∫x secxtanx+secxdxIntegrate by parts:  ∫fg'=fg-∫f'gf=x , g'=secxtanx+secxf'=1 and g=-1tanx+secx=-xtanx+secx-∫1-tanx-secxdxConsider, ∫1-tanx-secxdx=-∫1tanx+secxdxNow we solve : ∫1tanx+secxdxUse tanx=sinxcosx and secx=1cosx=∫cosx×1sinx+1dxPut u=sinx ⇒dx=ducosx=∫1u+1duPut v=u+1 ⇒du=dv=∫1vdv=lnv=lnu+1=lnsinx+1So, -∫1tanx+secxdx=-lnsinx+1So,-xtanx+secx-∫1-tanx-secxdx=lnsinx+1-xsecx+tanxNow we will put the limits
=lnsin3π4+1-3π4sec3π4+tan3π4-lnsinπ4+1-π4secπ4+tanπ4=ln12+1-3π4-2+-1-ln12+1-π42+1=ln1+22+3π42+1-ln1+22-π42+1=π2+1=π2+1×2-12-1=π2-12-1=π2-1
Regards

Q. The value of integral ∫ π / 4 3 π / 4 x 1 + sin x d x i s : ( a ) π 2 ( 2 + 1 ) ( b ) π 2 ( c ) 2 π ( 2 - 1 ) ( d ) π ( 2 - 1 )

Q. The value of integral $\int_{π / 4}^{3 π / 4} \frac{x}{1 + \sin x} d x i s : (a) \frac{π}{2} (\sqrt{2} + 1) (b) π \sqrt{2} (c) 2 π (\sqrt{2} - 1) (d) π (\sqrt{2} - 1)$