² x ² x \, dx = . . . . . . + C

Correct Answer is : tan⁡x−cot⁡x x - xtanx−cotx

Correct Answer is : tan⁡x−cot⁡x x - xtanx−cotx

Test Index

GUJCET 2026 Math Paper

Question : 6 of 40

Marks: +1, -0

\int \sec^2 x \csc^2 x \, dx =

+ C

Solution:

Concept:

Use trigonometric identities to simplify the product before integrating.

Explanation:

Let

I = \int \sec^2 x \csc^2 x \, dx

Write

\sec^2 x = \frac{1}{\cos^2 x}

and

\csc^2 x = \frac{1}{\sin^2 x}

I = \int \frac{1}{\sin^2 x \cos^2 x} \, dx

Use

1 = \sin^2 x + \cos^2 x

Thus

I = \int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} \, dx

Split the fraction:

I = \int \left( \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} \right) dx

I = \int (\sec^2 x + \csc^2 x) \, dx

Since

\int \sec^2 x \, dx = \tan x

and

\int \csc^2 x \, dx = -\cot x

Therefore

I = \tan x - \cot x + C

Answer:

\tan x - \cot x + C

, which matches option A.

Go to Question:

Prev Question

Next Question