Question

Evaluate: \[ \int e^{\sin x} \cdot \frac{x \cos^3 x - \sin x}{\cos^2 x} \, dx \]

• A. \( e^{\sin x}(x - \sec x) + C \)
• B. \( e^{\sin x}(x - \csc x) + C \)
• C. \( e^{\sin x}(x + \sec x) + C \)
• D. \( e^{\sin x}(x + \csc x) + C \)

Hint:

When integrating products with exponentials and trigonometric functions, look for patterns or use substitution.

Answer 1

The Correct Option is A

Let \( I = \int e^{\sin x} \cdot \frac{x \cos^3 x - \sin x}{\cos^2 x} \, dx \). Rewrite the integrand: \[ \frac{x \cos^3 x}{\cos^2 x} - \frac{\sin x}{\cos^2 x} = x \cos x - \tan x \cdot \sin x \] Then let us use substitution: Let \( u = \sin x \Rightarrow du = \cos x dx \). Further, recognizing the derivative of \( x \cos x \) terms, the expression matches: \[ \frac{d}{dx} \left( e^{\sin x} (x - \sec x) \right) \] Hence the result is: \[ e^{\sin x}(x - \sec x) + C \]