Question

Match List I with List II

List I		List II
A.	\(\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{7}{2}}x}{\sin^{\frac{7}{2}}+\cos^{\frac{7}{2}}}dx\)	I.	\(\frac{\pi}{4}-\frac{1}{2}\)
B.	\(\int\limits_0^{\pi}\frac{x\sin x}{1+\cos^2x}dx\)	II.	0
C.	\(\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}x\cos x\ dx\)	III.	\(\frac{\pi}{4}\)
D.	\(\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^2x\ dx\)	IV.	\(\frac{\pi^2}{4}\)

Choose the correct answer from the options given below :

• A. A-III, B-IV, C-II, D-I
• B. A-I, B-IV, C-II, D-III
• C. A-I, B-IV, C-II, D-III
• D. A-IV, B-III, C-II, D-I

Answer 1

The Correct Option is A

To solve the problem of matching List I with List II, we evaluate each integral and match it with the corresponding value.

List I		List II
A.	\(\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{7}{2}}x}{\sin^{\frac{7}{2}}x+\cos^{\frac{7}{2}}x}dx\)	I.	\(\frac{\pi}{4}-\frac{1}{2}\)
B.	\(\int\limits_0^{\pi}\frac{x\sin x}{1+\cos^2x}dx\)	II.	0
C.	\(\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}x\cos x\ dx\)	III.	\(\frac{\pi}{4}\)
D.	\(\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^2x\ dx\)	IV.	\(\frac{\pi^2}{4}\)

1. Integral A: \(\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{7}{2}}x}{\sin^{\frac{7}{2}}x+\cos^{\frac{7}{2}}x}dx\) evaluates to \(\frac{\pi}{4}\) by symmetry and properties of definite integrals. Thus, A matches with III.
2. Integral B: \(\int\limits_0^{\pi}\frac{x\sin x}{1+\cos^2x}dx\) is evaluated using symmetry and properties, resulting in \(\frac{\pi^2}{4}\). Thus, B matches with IV.
3. Integral C: \(\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}x\cos x\ dx\) calculates to 0 as it is an odd function over a symmetric interval. Thus, C matches with II.
4. Integral D: \(\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^2x\ dx\) results in \(\frac{\pi}{4}-\frac{1}{2}\) by trigonometric identities and integration. Thus, D matches with I.

The correct matching is A-III, B-IV, C-II, D-I.