Question

Find the integral of \(\frac{cos^2⁡x-sin^2⁡x}{7 cos^2⁡x sin^2⁡x}\).

(a) –\(\frac{1}{7}\) (cot⁡x-tan⁡x)+C

(b) –\(\frac{1}{7}\) (cot⁡x-2 tan⁡x)+C

(c) –\(\frac{1}{7}\) (cot⁡x+tan⁡x)+C

(d) –\(\frac{1}{7}\) (2 cot⁡x+3 tan⁡x)+C

The question was posed to me during an interview.

I'm obligated to ask this question of Methods of Integration-2 topic in portion Integrals of Mathematics – Class 12

Answer 1

Right choice is (c) –\(\frac{1}{7}\) (cot⁡x+tan⁡x)+C

Explanation: To find: \(\int \frac{cos^2⁡x-sin^2⁡x}{7 \,cos^2⁡x \,sin^2⁡x} dx\)

\(\int \frac{cos^2⁡x-sin^2⁡x}{7 \,cos^2⁡x \,sin^2⁡x} dx=\frac{1}{7} \int \frac{1}{sin^2⁡x}-\frac{1}{cos^2⁡x} dx\)

=\(\frac{1}{7} \int cosec^2 x-sec^2⁡x dx\)

=\(\frac{1}{7}\) (-cot⁡x-tan⁡x)+C

=-\(\frac{1}{7}\) (cot⁡x+tan⁡x)+C.