Calculus Examples

$x^{3} e^{x^{2}}$

Step 1

Write $x^{3} e^{x^{2}}$ as a function.

$f (x) = x^{3} e^{x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{3} e^{x^{2}} d x$

Step 4

Let $u = x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int u e^{u} \frac{1}{2} d u$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $u$ .

$\int \frac{u}{2} e^{u} d u$

Step 5.2

Combine $\frac{u}{2}$ and $e^{u}$ .

$\int \frac{u e^{u}}{2} d u$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int u e^{u} d u$

Step 7

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = e^{u}$ .

$\frac{1}{2} (u e^{u} - \int e^{u} d u)$

Step 8

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{1}{2} (u e^{u} - (e^{u} + C))$

Step 9

Simplify.

$\frac{1}{2} (u e^{u} - e^{u}) + C$

Step 10

Replace all occurrences of $u$ with $x^{2}$ .

$\frac{1}{2} (x^{2} e^{x^{2}} - e^{x^{2}}) + C$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{1}{2} (x^{2} e^{x^{2}}) + \frac{1}{2} (- e^{x^{2}}) + C$

Step 11.2

Multiply $\frac{1}{2} (x^{2} e^{x^{2}})$ .

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Step 11.2.1

Combine $x^{2}$ and $\frac{1}{2}$ .

$\frac{x^{2}}{2} e^{x^{2}} + \frac{1}{2} (- e^{x^{2}}) + C$

Step 11.2.2

Combine $\frac{x^{2}}{2}$ and $e^{x^{2}}$ .

$\frac{x^{2} e^{x^{2}}}{2} + \frac{1}{2} (- e^{x^{2}}) + C$

Step 11.3

Combine $\frac{1}{2}$ and $e^{x^{2}}$ .

$\frac{x^{2} e^{x^{2}}}{2} - \frac{e^{x^{2}}}{2} + C$

Step 12

Reorder terms.

$\frac{1}{2} x^{2} e^{x^{2}} - \frac{1}{2} e^{x^{2}} + C$

Step 13

The answer is the antiderivative of the function $f (x) = x^{3} e^{x^{2}}$ .

$F (x) =$ $\frac{1}{2} x^{2} e^{x^{2}} - \frac{1}{2} e^{x^{2}} + C$