Integration of function x e^ x^2 with respect to x is equal to : - Vidyadip

MCQ

Integration of function $\frac{x}{e^{x^2}}$ with respect to $x$ is equal to :

A
$\frac{1}{2 e^{x^2}}+ C$
B
$\frac{2}{e^{x^2}}+ C$
C
$\frac{-2}{e^{x^2}}+ C$
✓
$\frac{-1}{2 e^{x^2}}+ C$

✓

Answer

Correct option: D.

$\frac{-1}{2 e^{x^2}}+ C$

(D)
$\int \frac{x}{e^{x^2}} d x$
Let
$x^2=t \quad \therefore 2 x d x=d t$
$\Rightarrow \quad x d x=\frac{1}{2} d t$
$
\begin{aligned}
\int \frac{\frac{1}{2} d t}{e^t} & =\frac{1}{2} \int e^{-t} d t \\
& =\frac{-1}{2} e^{-t}+C \\
& =\frac{-1}{2 e^t}+C=\frac{-1}{2 e^{x^2}}+C
\end{aligned}
$
Hence correct option is (D).

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