Dear student ∫0πesin2x cos3x dxConsider , ∫esin2x cos3x dxPut u=sinx=∫eu21-u2du=∫eu2-eu2 u2 du=∫eu2 du-∫eu2 u2 duUse the common integral ∫eu2 du=eu2 Fu=eu2 Fu-∫eu2 u2 du=eu2 Fu-∫eu2 u u duApply integration by parts : u=u, v'=eu2 u=eu2 Fu-12eu2 u-∫12eu2 du =eu2 Fu-12eu2 u-12eu2 Fu =esin2x Fsinx-12esin2x sinx-12esin2x Fsinx =-12esin2x sinx+32esin2x Fsinx+CComputing the boundaries∫0πesin2x cos3x dx=-12esin2π sinπ+32esin2π Fsinπ--12esin20 sin0+32esin20 Fsin0=32F0-32F0=0 Regards

Please solve Q7

Please solve Q7 If f and g are continuous functions on [O, al such that f(x) = — x) and g(x) + g(a — x) = 2, then show that The value of eS*Xcos3x dx is The value of f cos5 xdx is equal to (D) none of (D) none of th cosx . In(x+ 1 x2 ) dx is equal to The value of J

Dear student

\int_{0}^{π} e^{\sin^{2} x} \cos^{3} x d x Consider, \int e^{\sin^{2} x} \cos^{3} x d x Put u = sinx = \int e^{u^{2}} (1 - u^{2}) du = \int (e^{u^{2}} - e^{u^{2}} u^{2}) du = \int e^{u^{2}} du - \int e^{u^{2}} u^{2} du Use the common integral \int e^{u^{2}} du = e^{u^{2}} F (u) = e^{u^{2}} F (u) - \int e^{u^{2}} u^{2} du = e^{u^{2}} F (u) - \int e^{u^{2}} u u du Apply integration by parts : u = u, v' = e^{u^{2}} u = e^{u^{2}} F (u) - (\frac{1}{2} e^{u^{2}} u - \int \frac{1}{2} e^{u^{2}} du) = e^{u^{2}} F (u) - (\frac{1}{2} e^{u^{2}} u - \frac{1}{2} e^{u^{2}} F (u)) = e^{\sin^{2} x} F (sinx) - (\frac{1}{2} e^{\sin^{2} x} sinx - \frac{1}{2} e^{\sin^{2} x} F (sinx)) = \frac{- 1}{2} e^{\sin^{2} x} sinx + \frac{3}{2} e^{\sin^{2} x} F (sinx) + C Computing the boundaries \int_{0}^{π} e^{\sin^{2} x} \cos^{3} x d x = [\frac{- 1}{2} e^{\sin^{2} π} sinπ + \frac{3}{2} e^{\sin^{2} π} F (sinπ)] - [\frac{- 1}{2} e^{\sin^{2} 0} \sin 0 + \frac{3}{2} e^{\sin^{2} 0} F (\sin 0)] = \frac{3}{2} F (0) - \frac{3}{2} F (0) = 0

Regards

View Full Answer

Please solve Q7 Please solve Q7 If f and g are continuous functions on [O, al such that f(x) = — x) and g(x) + g(a — x) = 2, then show that The value of eS*Xcos3x dx is The value of f cos5 xdx is equal to (D) none of (D) none of th cosx . In(x+ 1 x2 ) dx is equal to The value of J

Please solve Q7

Please solve Q7 If f and g are continuous functions on [O, al such that f(x) = — x) and g(x) + g(a — x) = 2, then show that The value of eS*Xcos3x dx is The value of f cos5 xdx is equal to (D) none of (D) none of th cosx . In(x+ 1 x2 ) dx is equal to The value of J