Question

Match the following.

	List I		List II
(I)	$\int^{1}_{-1} x |x |dx $	(a)	$\frac{\pi}{2}$
(II)	$\int^{\frac{\pi}{2}}_{0} \left( 1 + \log\left(\frac{4+3 \sin x }{4+3 \cos x}\right)\right)dx $	(b)	$\int^{\frac{\pi}{2}}_{0} f\left(x\right)dx $
(III)	$\int^{a}_{0} f(x) dx$	(c)	$\int^a_0 [ f(x) + f(-x)] dx $
(IV)	$\int^a_{-a} f(x) dx $	(d)	0
		(e)	$\int^a_0 f(a -x)dx$

• A. I $\to$ d, II $\to$ a, III $\to$ e , IV $\to$ c
• B. I $\to$ d, II $\to$ a, III $\to$ c , IV $\to$ b
• C. I $\to$ d, II $\to$ c, III $\to$ a , IV $\to$ e
• D. I $\to$ a, II $\to$ d, III $\to$ b , IV $\to$ d

Answer 1

Answer: (A)

$\int_\limits{-1}^{1} x|x| d x=0 [\because x|x|$ is an odd function]
II. Let $I=\int_\limits{0}^{\pi / 2}\left[1+\left(\log \frac{4+3 \sin x}{4+3 \cos x}\right)\right] d x$
$\left.I=\int_{0}^{\pi / 2}\left(1+\mid \log \frac{4+3 \sin \left(\frac{\pi}{2}-x\right)}{4+3 \cos \left(\frac{\pi}{2}-x\right)}\right]\right) d x$
$I=\int_{0}^{\pi / 2}\left(1+\left(\log \frac{4+3 \cos x}{4+3 \sin x}\right)\right) d x$
$\Rightarrow 2 I=\int\limits_{0}^{\pi / 2}\left(2+\log \frac{4+3 \sin x}{4+3 \cos x}\right.$
$\left.+\log \frac{4+3 \cos x}{4+3 \sin x}\right) d x$
$\Rightarrow 2 I=\int\limits_{0}^{\pi / 2}\left(2+\log \frac{(4+3 \sin x)(4+3 \cos x)}{(4+3 \cos x)(4+3 \sin x)}\right) d x$
$\Rightarrow 2 I=\int_{0}^{\pi / 2}(2+\log 1) d x=\int_\limits{0}^{\pi / 2} \,2 \,d x=\pi$
$\Rightarrow I=\frac{\pi}{2}$
III. $\int_\limits{0}^{a} f(x) \,d x=\int_\limits{0}^{a} f(a-x)\, d x$
IV. $ \int_\limits{-a}^{a} f(x) \,d x =\int_\limits{0}^{a} f(x) \,d x+\int_\limits{0}^{a} f(-x) \,d x $
$ =\int_\limits{0}^{a}[f(x)+f(-x)] \,d x $