Question

$\int e^{x^4}(x+x^3+2x^5)e^{x^2}dx$ is equal to

• A. $\frac{1}{2}x{e}^{x^2}{e}^{x^4}+c$
• B. $\frac{1}{2}{x}^2{e}^{x^4}+c$
• C. $\frac{1}{2}{e}^{x^2}{e}^{x^4}+c$
• D. $\frac{1}{2}{x}^2{e}^{x^2}{e}^{x^4}+c$

Answer 1

The correct option is D $\frac{1}{2}{x}^2{e}^{x^2}{e}^{x^4}+c$

Putting $x^2=t$
$\Rightarrow 2xdx=dt$
$\Rightarrow I=\frac{1}{2}\int e^{t^2}(1+t+2t^2)e^{t}dt=\frac{1}{2}\int e^t[t{e}^{t^2}+(e^{t^2}+2t^2{e}^{t^2})]dt\underset{f\left(t\right)}{\downarrow }\underset{f^{\prime }\left(t\right)}{\downarrow }=\frac{1}{2}\int e^t\left[f\right(t)+f^{\prime }(t\left)\right]dt=\frac{1}{2}{e}^t\left(t{e}^{t^2}\right)+c$
$\Rightarrow I=\frac{1}{2}{x}^2{e}^{x^2}{e}^{x^4}+c$