Question

Show that$F\left(x\right)=\frac{1}{2}{e}^{\left(x^2\right)}\left(x^2-1\right)$is an anti-derivative of$f\left(x\right)=x^3{e}^{\left(x^2\right)}.$

Short answer

It is shown that$F\left(x\right)=\frac{1}{2}{e}^{\left(x^2\right)}\left(x^2-1\right)\text{is an anti-derivative of}f\left(x\right)=x^3{e}^{\left(x^2\right)}$.

Step-by-step solution

Step 1. Given Information.

The anti derivative is$F\left(x\right)=\frac{1}{2}{e}^{\left(x^2\right)}\left(x^2-1\right)\text{of}f\left(x\right)=x^3{e}^{\left(x^2\right)}$.

Step 2. Conclusion.

On differentiating the function, we get,

$$\begin{gathered} \frac{d}{dx}{x}^3{e}^{\left(x^2\right)} \\ =2x^2{e}^{\left(x^2\right)}-e^{\left(x^2\right)} \\ =2e^{\left(x^2\right)}\left(x^2-1\right) \\ So, \\ \int x^3{e}^{\left(x^2\right)}dx \\ =\frac{1}{2}{e}^{\left(x^2\right)}\left(x^2-1\right)+C \end{gathered}$$

Therefore,

$F\left(x\right)=\frac{1}{2}{e}^{\left(x^2\right)}\left(x^2-1\right)\text{is an anti-derivative of}f\left(x\right)=x^3{e}^{\left(x^2\right)}.$