Question

Evaluate $\int x{e}^{x^2}(\sin x^2+\cos x^2)dx$

• A. $\frac{1}{2}{e}^{x^2}\cos \left(x^2\right)+c$
• B. $\frac{1}{2}{e}^{-x^2}\cos \left(x^2\right)+c$
• C. $\frac{1}{2}{e}^{-x^2}\sin \left(x^2\right)+c$
• D. $\frac{1}{2}{e}^{x^2}\sin \left(x^2\right)+c$

Answer 1

The correct option is D $\frac{1}{2}{e}^{x^2}\sin \left(x^2\right)+c$

Let $I=\int x{e}^{x^2}(\sin x^2+\cos x^2)dx$
Put $e^{x^2}=t\Rightarrow 2x{e}^{x^2}dx=dt$
$I=\frac{1}{2}\int (\sin \log t+\cos \log t)dt$
$=\frac{1}{2}\left[t\sin \log t\right]-\frac{1}{2}\int \cos \log tdt+\frac{1}{2}\int \cos \log tdt$
$=\frac{1}{2}{e}^{x^2}\sin x^2+c$