Calculus Examples

$\int x^{11} e^{x^{12}} d x$

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 $d u_{1}$ .

$\int {\sqrt{u_{1}}}^{10} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u_{1}}}^{10}$ as ${u_{1}}^{5}$ .

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Step 2.1.1

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

$\int {({u_{1}}^{\frac{1}{2}})}^{10} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.2

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

$\int {u_{1}}^{\frac{1}{2} \cdot 10} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.3

Combine $\frac{1}{2}$ and $10$ .

$\int {u_{1}}^{\frac{10}{2}} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.4

Cancel the common factor of $10$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $10$ .

$\int {u_{1}}^{\frac{2 \cdot 5}{2}} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int {u_{1}}^{\frac{2 \cdot 5}{2 (1)}} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.2

Cancel the common factor.

$\int {u_{1}}^{\frac{2 \cdot 5}{2 \cdot 1}} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.3

Rewrite the expression.

$\int {u_{1}}^{\frac{5}{1}} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.4

Divide $5$ by $1$ .

$\int {u_{1}}^{5} e^{{\sqrt{u_{1}}}^{12}} \frac{1}{2} d u_{1}$

Step 2.2

Rewrite ${\sqrt{u_{1}}}^{12}$ as ${u_{1}}^{6}$ .

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Step 2.2.1

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

$\int {u_{1}}^{5} e^{{({u_{1}}^{\frac{1}{2}})}^{12}} \frac{1}{2} d u_{1}$

Step 2.2.2

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

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{1}{2} \cdot 12}} \frac{1}{2} d u_{1}$

Step 2.2.3

Combine $\frac{1}{2}$ and $12$ .

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{12}{2}}} \frac{1}{2} d u_{1}$

Step 2.2.4

Cancel the common factor of $12$ and $2$ .

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Step 2.2.4.1

Factor $2$ out of $12$ .

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{2 \cdot 6}{2}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $2$ out of $2$ .

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{2 \cdot 6}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.2

Cancel the common factor.

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{2 \cdot 6}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.3

Rewrite the expression.

$\int {u_{1}}^{5} e^{{u_{1}}^{\frac{6}{1}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.4

Divide $6$ by $1$ .

$\int {u_{1}}^{5} e^{{u_{1}}^{6}} \frac{1}{2} d u_{1}$

Step 2.3

Combine $\frac{1}{2}$ and ${u_{1}}^{5}$ .

$\int \frac{{u_{1}}^{5}}{2} e^{{u_{1}}^{6}} d u_{1}$

Step 2.4

Combine $\frac{{u_{1}}^{5}}{2}$ and $e^{{u_{1}}^{6}}$ .

$\int \frac{{u_{1}}^{5} e^{{u_{1}}^{6}}}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int {u_{1}}^{5} e^{{u_{1}}^{6}} d u_{1}$

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

$2 u_{1}$

Step 4.2

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

$\frac{1}{2} \int {\sqrt{u_{2}}}^{4} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

$\frac{1}{2} \int {({u_{2}}^{\frac{1}{2}})}^{4} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.2

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

$\frac{1}{2} \int {u_{2}}^{\frac{1}{2} \cdot 4} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.3

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

$\frac{1}{2} \int {u_{2}}^{\frac{4}{2}} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.4

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

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Step 5.1.4.1

Factor $2$ out of $4$ .

$\frac{1}{2} \int {u_{2}}^{\frac{2 \cdot 2}{2}} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.4.2

Cancel the common factors.

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Step 5.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \int {u_{2}}^{\frac{2 \cdot 2}{2 (1)}} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.2

Cancel the common factor.

$\frac{1}{2} \int {u_{2}}^{\frac{2 \cdot 2}{2 \cdot 1}} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.3

Rewrite the expression.

$\frac{1}{2} \int {u_{2}}^{\frac{2}{1}} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} \int {u_{2}}^{2} e^{{\sqrt{u_{2}}}^{6}} \frac{1}{2} d u_{2}$

Step 5.2

Rewrite ${\sqrt{u_{2}}}^{6}$ as ${u_{2}}^{3}$ .

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Step 5.2.1

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

$\frac{1}{2} \int {u_{2}}^{2} e^{{({u_{2}}^{\frac{1}{2}})}^{6}} \frac{1}{2} d u_{2}$

Step 5.2.2

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

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{1}{2} \cdot 6}} \frac{1}{2} d u_{2}$

Step 5.2.3

Combine $\frac{1}{2}$ and $6$ .

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{6}{2}}} \frac{1}{2} d u_{2}$

Step 5.2.4

Cancel the common factor of $6$ and $2$ .

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Step 5.2.4.1

Factor $2$ out of $6$ .

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{2 \cdot 3}{2}}} \frac{1}{2} d u_{2}$

Step 5.2.4.2

Cancel the common factors.

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Step 5.2.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{2 \cdot 3}{2 (1)}}} \frac{1}{2} d u_{2}$

Step 5.2.4.2.2

Cancel the common factor.

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{2 \cdot 3}{2 \cdot 1}}} \frac{1}{2} d u_{2}$

Step 5.2.4.2.3

Rewrite the expression.

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{\frac{3}{1}}} \frac{1}{2} d u_{2}$

Step 5.2.4.2.4

Divide $3$ by $1$ .

$\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{3}} \frac{1}{2} d u_{2}$

Step 5.3

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

$\frac{1}{2} \int \frac{{u_{2}}^{2}}{2} e^{{u_{2}}^{3}} d u_{2}$

Step 5.4

Combine $\frac{{u_{2}}^{2}}{2}$ and $e^{{u_{2}}^{3}}$ .

$\frac{1}{2} \int \frac{{u_{2}}^{2} e^{{u_{2}}^{3}}}{2} d u_{2}$

Step 6

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

$\frac{1}{2} (\frac{1}{2} \int {u_{2}}^{2} e^{{u_{2}}^{3}} d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{2 \cdot 2} \int {u_{2}}^{2} e^{{u_{2}}^{3}} d u_{2}$

Step 7.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int {u_{2}}^{2} e^{{u_{2}}^{3}} d u_{2}$

Step 8

Let $u_{3} = {u_{2}}^{3}$ . Then $d u_{3} = 3 {u_{2}}^{2} d u_{2}$ , so $\frac{1}{3} d u_{3} = {u_{2}}^{2} d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 8.1

Let $u_{3} = {u_{2}}^{3}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 8.1.1

Differentiate ${u_{2}}^{3}$ .

$\frac{d}{d u_{2}} [{u_{2}}^{3}]$

Step 8.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$3 {u_{2}}^{2}$

Step 8.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{4} \int e^{u_{3}} \frac{1}{3} d u_{3}$

Step 9

Combine $e^{u_{3}}$ and $\frac{1}{3}$ .

$\frac{1}{4} \int \frac{e^{u_{3}}}{3} d u_{3}$

Step 10

Since $\frac{1}{3}$ is constant with respect to $u_{3}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{4} (\frac{1}{3} \int e^{u_{3}} d u_{3})$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{3}$ by $\frac{1}{4}$ .

$\frac{1}{3 \cdot 4} \int e^{u_{3}} d u_{3}$

Step 11.2

Multiply $3$ by $4$ .

$\frac{1}{12} \int e^{u_{3}} d u_{3}$

Step 12

The integral of $e^{u_{3}}$ with respect to $u_{3}$ is $e^{u_{3}}$ .

$\frac{1}{12} (e^{u_{3}} + C)$

Step 13

Simplify.

$\frac{1}{12} e^{u_{3}} + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{3}$ with ${u_{2}}^{3}$ .

$\frac{1}{12} e^{{u_{2}}^{3}} + C$

Step 14.2

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

$\frac{1}{12} e^{{({u_{1}}^{2})}^{3}} + C$

Step 14.3

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

$\frac{1}{12} e^{{({(x^{2})}^{2})}^{3}} + C$