Q. 13 - Calculus

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 $f\left(x\right)=x^3{e}^{\left(x^2\right)}.$

Step-by-Step Solution

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 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,

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