Question

Evaluate the integral: \[ \int \frac{(\sin^4x + 2\cos^2x - 1) \cos x}{(1 + \sin x)^6} \, dx \]

• A. \( \frac{\sin^6 x}{6(1 + \sin x)^6} + c \)
• B. \( - \frac{\sin^6 x}{6(1 + \sin x)^6} + c \)
• C. \( \frac{\cos^6 x}{6(1 + \sin x)^6} + c \)
• D. \( - \frac{\cos^6 x}{6(1 + \sin x)^6} + c \) \bigskip

Hint:

For trigonometric integrals, substitution is a powerful tool. Here, we used \( u = 1 + \sin x \) to simplify the expression and handle the powers of sine and cosine functions.

Answer 1

The Correct Option is D

Step 1: Notice that the structure of the integral suggests using substitution and simplifying the expression. Let \[ u = 1 + \sin x, \quad du = \cos x \, dx. \] Rewriting the integral with respect to \( u \), we have: \[ \int \frac{(\sin^4 x + 2\cos^2x - 1) \cos x}{(1 + \sin x)^6} \, dx = - \frac{\cos^6 x}{6(1 + \sin x)^6} + c. \] \bigskip