Question

\(\int \sin^3 x \, e^{\log \cos x} \, dx =\)

• A. $\frac{\cos^4 x}{4} + C$
• B. $-\frac{\cos^4 x}{4} + C$
• C. $\frac{x\cos^4 x}{4} + C$
• D. $\frac{\sin^4 x}{4} + C$
• E. $-\frac{\sin^4 x}{4} + C$

Hint:

Whenever you see $e^{\log(\text{something})}$: - Directly simplify it to that expression - Then apply substitution if needed

Answer 1

The Correct Option is D

Concept: Use the identity: \[ e^{\log a} = a \] Then simplify the integrand and use substitution.

Step 1: Simplify $e^{\log \cos x}$.
\[ e^{\log \cos x} = \cos x \]

Step 2: Rewrite the integral.
\[ \int \sin^3 x \cdot \cos x \, dx \]

Step 3: Use substitution.
Let: \[ u = \sin x \quad \Rightarrow \quad du = \cos x \, dx \]

Step 4: Transform the integral.
\[ \int u^3 \, du \]

Step 5: Integrate.
\[ \frac{u^4}{4} + C \]

Step 6: Substitute back.
\[ = \frac{\sin^4 x}{4} + C \]