Question

Evaluate the following integral:
$\int {x{e^{{x^2}}}dx}$

Answer 1

Hint: This problem can be solved by a substitution method. The substitution method (also called u-substitution) is used when an integral contains some function and its derivative. In this case, we can set u equal to the function and rewrite the integral in terms of new variable u. This makes integral easier to solve. So, use this concept to reach the solution of the problem.

Complete step-by-step answer:
Let $I = \int {x{e^{{x^2}}}dx}$
Put ${x^2} = t$ and differentiate it w.r.t ‘$x$’

$$\Rightarrow \dfrac{{d\left( {{x^2}} \right)}}{{dx}} = \dfrac{{d\left( t \right)}}{{dx}} \\\ \Rightarrow 2x = \dfrac{{dt}}{{dx}} \\\ \Rightarrow 2xdx = dt \\\ \therefore xdx = \dfrac{{dt}}{2} \\\$$

Then $I = \int {{e^t}\dfrac{{dt}}{2}}$
$\Rightarrow I = \dfrac{1}{2}\int {{e^t}dt}$
We know that $\int {{e^x}dx} = {e^x} + c$

$$\Rightarrow I = \dfrac{1}{2}\left( {{e^t} + c} \right) \\\ \Rightarrow I = \dfrac{1}{2}{e^t} + \dfrac{c}{2} \\\ \therefore I = \dfrac{1}{2}{e^t} + C \\\$$

Since, ${x^2} = t$
Therefore, $I = \dfrac{1}{2}{e^{{x^2}}} + C$
Thus, $\int {x{e^{{x^2}}}dx} = \dfrac{1}{2}{e^{{x^2}}} + C$

Note: A constant namely integrating constant that is added to the function obtained by evaluating the indefinite integral of a given function, indicating that all indefinite integrals of the given function differ by, at most, a constant. So, it is necessary to add integrating constants after completion of the integral.