Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f, g$ and $h$ be continuous functions on $[0,a]$ such that $f(x) = f(a-x), g(x) = -g(a-x)$ and $3h(x) - 4h(a-x) = 5$. Then $\int_0^a f(x)g(x)h(x)dx =$
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