# Calculate the following antiderivative. Show your work. \int xe^{x^2} dx

## Question:

Calculate the following antiderivative. Show your work. {eq}\int xe^{x^2} dx {/eq}

## Antiderivative of a Function:

The antiderivative of {eq}\displaystyle f(x) {/eq} is {eq}F(x) = \int f(x) \ dx {/eq}.

To compute the indefinite integral, we'll take the help of substitution. Integration by substitution is a process of substituting the values of an integral temporarily. The value of indefinite integral is not fixed, we need to add constant C after integrating.

We are given:

{eq}\displaystyle \int xe^{x^{2}}dx {/eq}

Apply u-substitution {eq}u= x^2 \rightarrow \ du = 2x \ dx {/eq}

{eq}= \displaystyle \int \dfrac{1}{2} e^{u} du {/eq}

Take the constant out:

{eq}= \displaystyle \dfrac{1}{2} \int e^{u} du {/eq}

{eq}= \displaystyle \dfrac{1}{2} e^{ u}+C {/eq}

Substitute back {eq}u= x^2 {/eq}

{eq}= \displaystyle \dfrac{1}{2} e^{x^2}+C {/eq}

Therefore, the solution is:

{eq}\displaystyle {\boxed{\int xe^{x^2} dx = \displaystyle \dfrac{1}{2} e^{x^2}+C.}} {/eq}