Indefinite Integration
e^(sin²x) Type Integral Using f+f' Form
nta_pyq_2023_apr
Grade 12
Question:
If $I(x)=\displaystyle\int e^{\sin^2x}(\cos x\sin 2x-\sin x)\,dx$ and $I(0)=1$, then $I\!\left(\dfrac{\pi}{3}\right)$ is equal to
$-\dfrac{1}{2}e^{3/4}$
$\dfrac{1}{2}e^{3/4}$
$-e^{3/4}$
$e^{3/4}$
Step-by-Step Solution
Key Concept: Write integrand as $e^{\sin^2x}(\cos x-\frac{1}{2\cos x})\sin2x\,dx$. Substitute $t=\sin^2x$: $I=\int e^t(\sqrt{1-t}-\frac{1}{2\sqrt{1-t}})dt=e^t\sqrt{1-t}+C$ (using $\int e^t(f+f')dt=e^tf+C$).
$I(x)=e^{\sin^2x}\cos x$. $I(\frac{\pi}{3})=\frac{1}{2}e^{3/4}$.
Correct Answer: 2