#### Provide Solution For R.D.Sharma Maths Class 12 Chapter 19 definite Integrals Exercise 19.1 Question 30 Maths Textbook Solution.

Answer: $\frac{-\pi }{4}$

Hint: Use indefinite formula then put the limit to solve this integral

Given: $\int_{0}^{\frac{\pi}{2}} x^{2} \cos 2 x d x$

Solution:

$\int_{0}^{\frac{\pi}{2}} x^{2} \cos 2 x d x$

Integrating by parts then

\begin{aligned} &=\left[x^{2} \int \cos 2 x d x\right]_{0}^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}}\left\{\frac{d\left(x^{2}\right)}{d x} \int \cos 2 x d x\right\} d x \\ &=\left[x^{2} \frac{\sin 2 x}{2}\right]_{0}^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}} 2 x \frac{\sin 2 x}{2} \end{aligned} \quad\left[\int \cos a x d x=\frac{\sin a x}{a}\right]

$=\frac{1}{2}\left[x^{2} \sin 2 x\right]-\frac{2}{2} \int_{0}^{\frac{\pi}{2}} x \sin 2 x d x$

Again using integrating on by parts method

$=\frac{1}{2}\left[\left(\frac{\pi}{2}\right)^{2} \sin 2 \times \frac{\pi}{2}-0^{2} \sin 0\right]-\left[x \int \sin 2 x d x\right]_{0}^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}}\left(\frac{d(x)}{d x} \int \sin 2 x d x\right) d x$

\begin{aligned} &=\frac{1}{2}\left[\frac{\pi^{2}}{4} \sin \pi-\sin 0 \times 0\right]-\left[x\left(\frac{-\cos 2 x}{2}\right)\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\left(\frac{-\cos 2 x}{2}\right) d x \\ &=\frac{1}{2} \times 0+\frac{1}{2}[x \cos 2 x]_{0}^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}} \frac{\cos 2 x}{2} d x & {\left[\begin{array}{l} \int \sin a x d x=\frac{-\cos a x}{a} \\ \sin \pi=0, \sin 0=0 \end{array}\right]} \end{aligned}

$=\frac{1}{2}\left[\frac{\pi}{2} \cos 2 \times \frac{\pi}{2}-0 \times \cos 2 \times 0\right]-\frac{1}{2}\left[\frac{\sin 2 x}{2}\right]_{0}^{\frac{\pi}{2}} \quad\left[\int \cos a x d x=\frac{\sin a x}{a}\right]$

\begin{aligned} &=\frac{1}{2}\left[\frac{\pi}{2} \cos \pi-0\right]-\frac{1}{2 \times 2}[\sin 2 x]_{0}^{\frac{\pi}{2}} \\ &=\frac{1}{2}\left[\frac{\pi}{2}\right](-1)-\frac{1}{4}\left[\sin 2 \times \frac{\pi}{2}-\sin 2 \times 0\right] \end{aligned}                                                            $\left [ \cos \pi =-1 \right ]$

\begin{aligned} &=\frac{1}{2}\left(\frac{-\pi}{2}\right)-\frac{1}{4}[\sin \pi-\sin 0] \\ &=-\frac{\pi}{4}-\frac{1}{4}[0-0] \end{aligned}                                                                            $\left [ \sin \pi =\sin 0=0 \right ]$

$=\frac{-\pi }{4}$