Question

Solve the integral:$\int 3x{e}^{x^2}dx$

Short answer

The required answer is$\frac{3}{2}{e}^{x^2}+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int 3x{e}^{x^2}dx$.

Step 2. Solve the integration by parts . 

We have,

$$\begin{gathered} u=x^2 \\ du=2xdx \\ \frac{du}{2}=xdx \end{gathered}$$

The formula of integration by parts is $\int udv=uv-\int vdu$

$$\begin{gathered} \int 3x{e}^{x^2}dx \\ =3\int x{e}^{x^2}dx \\ =3\left[\frac{1}{2}\int e^{x^2}dx\right] \\ =\frac{3}{2}{e}^{x^2}+c \end{gathered}$$