Question

Evaluate \( \int \frac{\sec^2 x}{\sin^7 x} \, dx - \int \frac{7}{\sin^7 x} \, dx \):

• A. \( \frac{1}{\sin^6 x \cos x} + c \)
• B. \( \frac{\tan x}{\sin^8 x} + c \)
• C. \( \sin^8 x \cos x + c \)
• D. \( \sec x \tan^7 x + c \)

Hint:

Sometimes, rather than solving directly, differentiating the options can help verify the correct antiderivative. Use this when the integral is complicated or resembles a standard pattern.

Answer 1

The Correct Option is A

Given: \[ \int \frac{\sec^2 x}{\sin^7 x} \, dx - \int \frac{7}{\sin^7 x} \, dx = \int \left( \frac{\sec^2 x - 7}{\sin^7 x} \right) dx \] Now, check the derivative of: \[ f(x) = \frac{1}{\sin^6 x \cos x} \] Differentiate using the product and chain rules. We get: \[ \frac{d}{dx} \left( \frac{1}{\sin^6 x \cos x} \right) = \frac{\sec^2 x - 7}{\sin^7 x} \] Hence, \[ \int \left( \frac{\sec^2 x - 7}{\sin^7 x} \right) dx = \frac{1}{\sin^6 x \cos x} + c \]