Question

$\int_{}^{\int}{x\sin}$2x . cos 3x dx is –

• A. –$\frac{x\cos 5x}{10}$+ $\frac{\sin 5x}{50}$ + c
• B. – $\frac{x\cos 5x}{10}$+$\frac{\sin 5x}{50}$+$\frac{x\cos x}{2}$– $\frac{\sin x}{2}$+ c
• C. –$\frac{x\cos 5x}{10}$+$\frac{\sin 5x}{50}$–$\frac{x\cos x}{2}$+ $\frac{\sin x}{2}$+ c
• D. None of these

Answer 1

Answer: (B)

– $\frac{x\cos 5x}{10}$+$\frac{\sin 5x}{50}$+$\frac{x\cos x}{2}$– $\frac{\sin x}{2}$+ c

Answer 2

I = $\int_{}^{}x$sin 2x . cos 3x dx

= $\frac { 1 } { 2 }$ $\int_{}^{}{2x}$ cos 3x . sin 2x dx = $\frac { 1 } { 2 }$ $\int_{}^{}x$ (sin 5x – sin x) dx

= $\frac { 1 } { 2 }$ $\int_{}^{}x$ sin 5x dx– $\frac { 1 } { 2 }$ $\int_{}^{}x$sin x dx …(1)

Let I₁ =$\int_{}^{}{\underset{I}{\overset{x}{︸}}\underset{II}{\overset{\sin 5x}{︸}}}$dx = x $\left( - \frac{\cos 5x}{5} \right)$ –

$\int_{}^{}1$ . $\left( - \frac{\cos 5x}{5} \right)$dx

= – $\frac{x\cos 5x}{5}$ + $\frac{\sin 5x}{25}$

& I₂ = $\int_{}^{}{\underset{I}{\overset{x}{︸}}\underset{II}{\overset{\sin x}{︸}}}$dx = x(–cos x) –$\int_{}^{}1$ . (–cos x) dx

= –x cos x + sin x

Put I₁ & I₂ in equation (1)

Ž I = –$\frac{x\cos 5x}{10}$+$\frac{\sin 5x}{50}$+$\frac{x\cos x}{2}$– $\frac{\sin x}{2}$+ c