Calculus Examples

$\int x^{3} e^{4 x^{4}} d x$

Step 1

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

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Step 1.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

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 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int u_{1} e^{4 {\sqrt{u_{1}}}^{4}} \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

$\int u_{1} e^{4 {({u_{1}}^{\frac{1}{2}})}^{4}} \frac{1}{2} d u_{1}$

Step 2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int u_{1} e^{4 {u_{1}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{1}$

Step 2.1.3

Combine $\frac{1}{2}$ and $4$ .

$\int u_{1} e^{4 {u_{1}}^{\frac{4}{2}}} \frac{1}{2} d u_{1}$

Step 2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $4$ .

$\int u_{1} e^{4 {u_{1}}^{\frac{2 \cdot 2}{2}}} \frac{1}{2} d u_{1}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int u_{1} e^{4 {u_{1}}^{\frac{2 \cdot 2}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.2

Cancel the common factor.

$\int u_{1} e^{4 {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.3

Rewrite the expression.

$\int u_{1} e^{4 {u_{1}}^{\frac{2}{1}}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.4

Divide $2$ by $1$ .

$\int u_{1} e^{4 {u_{1}}^{2}} \frac{1}{2} d u_{1}$

Step 2.2

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

$\int \frac{u_{1}}{2} e^{4 {u_{1}}^{2}} d u_{1}$

Step 2.3

Combine $\frac{u_{1}}{2}$ and $e^{4 {u_{1}}^{2}}$ .

$\int \frac{u_{1} e^{4 {u_{1}}^{2}}}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int u_{1} e^{4 {u_{1}}^{2}} d u_{1}$

Step 4

Let $u_{2} = 4 {u_{1}}^{2}$ . Then $d u_{2} = 8 u_{1} d u_{1}$ , so $\frac{1}{8} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.1

Let $u_{2} = 4 {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 4.1.1

Differentiate $4 {u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [4 {u_{1}}^{2}]$

Step 4.1.2

Since $4$ is constant with respect to $u_{1}$ , the derivative of $4 {u_{1}}^{2}$ with respect to $u_{1}$ is $4 \frac{d}{d u_{1}} [{u_{1}}^{2}]$ .

$4 \frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 4.1.3

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

$4 (2 u_{1})$

Step 4.1.4

Multiply $2$ by $4$ .

$8 u_{1}$

Step 4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} \int e^{u_{2}} \frac{1}{8} d u_{2}$

Step 5

Combine $e^{u_{2}}$ and $\frac{1}{8}$ .

$\frac{1}{2} \int \frac{e^{u_{2}}}{8} d u_{2}$

Step 6

Since $\frac{1}{8}$ is constant with respect to $u_{2}$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{2} (\frac{1}{8} \int e^{u_{2}} d u_{2})$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{8}$ by $\frac{1}{2}$ .

$\frac{1}{8 \cdot 2} \int e^{u_{2}} d u_{2}$

Step 7.2

Multiply $8$ by $2$ .

$\frac{1}{16} \int e^{u_{2}} d u_{2}$

Step 8

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{16} (e^{u_{2}} + C)$

Step 9

Simplify.

$\frac{1}{16} e^{u_{2}} + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

Replace all occurrences of $u_{2}$ with $4 {u_{1}}^{2}$ .

$\frac{1}{16} e^{4 {u_{1}}^{2}} + C$

Step 10.2

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$\frac{1}{16} e^{4 {(x^{2})}^{2}} + C$