Indefinite Integration
Integral Calculus-1
star_batch_jee_advanced_2025
Grade None
Question:
The value of the integral $$\int e^{\sin^2 x}(\cos x + \cos^3 x) \sin x dx$$ is:
\frac{1}{2}e^{\sin^2 x}(3 - \sin^2 x) + c
e^{\sin^2 x}\left(1 + \frac{1}{2}\cos^2 x\right) + c
e^{\sin^2 x}(3\cos^2 x + 2\sin^2 x) + c
e^{\sin^2 x}(2\cos^2 x + 3\sin^2 x) + c
Step-by-Step Solution
Key Concept: The substitution $t = \sin^2 x$ simplifies the exponential integral by eliminating trigonometric functions.
Substituting $t = \sin^2 x$ gives $dt = 2\sin x \cos x dx$. The integral reduces to $\frac{1}{2}\int (2-t) e^t dt = \frac{3}{2}e^t - \frac{te^t}{2} + C$. Substituting back yields $e^{\sin^2 x}\left(1 + \frac{\cos^2 x}{2}\right) + C$.
Correct Answer: 1,2