Question

The integral $I=\displaystyle \int \left[x e^{x^{2}} \left(sin x^{2} + cos ⁡ x^{2}\right)\right]dx$ $=f\left(x\right)+c,$ (where, $c$ is the constant of integration). Then, $f\left(x\right)$ can be

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

Answer 1

Answer: (D)

Let, $x^{2}=t\Rightarrow 2xdx=dt$
$\therefore I=\frac{1}{2}\displaystyle \int e^{t}\left(sin t + cos ⁡ t\right)dt$
$=\frac{1}{2}e^{t}\cdot sin t+c$
$=\frac{1}{2}e^{x^{2}}sin \left(x^{2}\right)+c$
$\left\{\right.$ As, $\left.\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} \cdot f(x)+c\right\}$