Integral de $$$x e^{x^{2}}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu aportación
Encuentra $$$\int x e^{x^{2}}\, dx$$$.
Solución
Let $$$u=x^{2}$$$.
Then $$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$ (steps can be seen here), and we have that $$$x dx = \frac{du}{2}$$$.
The integral becomes
$${\color{red}{\int{x e^{x^{2}} d x}}} = {\color{red}{\int{\frac{e^{u}}{2} d u}}}$$
Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = e^{u}$$$:
$${\color{red}{\int{\frac{e^{u}}{2} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{2}\right)}}$$
The integral of the exponential function is $$$\int{e^{u} d u} = e^{u}$$$:
$$\frac{{\color{red}{\int{e^{u} d u}}}}{2} = \frac{{\color{red}{e^{u}}}}{2}$$
Recall that $$$u=x^{2}$$$:
$$\frac{e^{{\color{red}{u}}}}{2} = \frac{e^{{\color{red}{x^{2}}}}}{2}$$
Therefore,
$$\int{x e^{x^{2}} d x} = \frac{e^{x^{2}}}{2}$$
Add the constant of integration:
$$\int{x e^{x^{2}} d x} = \frac{e^{x^{2}}}{2}+C$$
Answer: $$$\int{x e^{x^{2}} d x}=\frac{e^{x^{2}}}{2}+C$$$