Question

The value of $\int\limits_{-\pi / 2}^{\pi / 2}\left(x^3+x \cos x+\tan 5 x+1\right) d x$ is

• A. 0
• B. 2
• C. $\pi$
• D. 1

Answer 1

Answer: (C)

Let $I=\int\limits_{-\pi / 2}^{\pi / 2}\left(x^3+x \cos x+\tan ^5 x+1\right) d x$
$\Rightarrow I=\int\limits_{-\pi / 2}^{\pi / 2} x^3 d x+\int\limits_{-\pi / 2}^{\pi / 2} x \cos x d x $
$ +\int\limits_{-\pi / 2}^{\pi / 2} \tan ^5 x d x+\int\limits_{-\pi / 2}^{\pi / 2} 1 d x$
We know that,
$ \int\limits_{-a}^a f(x) d x=\begin{cases} 2 \int\limits_0^a f(x) d x, & \text { if } f(x) \text { is even } \\ 0, & \text { if } f(x) \text { is odd } \end{cases} $
$ \therefore I=0+0+0+2 \int\limits_0^{\pi / 2} 1 d x . $
${\left[\because x^3, x \cos x \text { and } \tan ^5(x)\right. \text { are odd functions] }}$
$ \therefore I=2[x]_0^{\pi / 2}=\frac{2 \pi}{2}=\pi $