Calculus Examples

$\int 24 t^{7} e^{- t^{8}} d t$

Step 1

Since $24$ is constant with respect to $t$ , move $24$ out of the integral.

$24 \int t^{7} e^{- t^{8}} d t$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $t^{2}$ .

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

Step 2.1.2

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

$2 t$

Step 2.2

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

$24 \int {\sqrt{u_{1}}}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Rewrite ${\sqrt{u_{1}}}^{6}$ as ${u_{1}}^{3}$ .

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Step 3.1.1

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

$24 \int {({u_{1}}^{\frac{1}{2}})}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.2

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

$24 \int {u_{1}}^{\frac{1}{2} \cdot 6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.3

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

$24 \int {u_{1}}^{\frac{6}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.4

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

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Step 3.1.4.1

Factor $2$ out of $6$ .

$24 \int {u_{1}}^{\frac{2 \cdot 3}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

$24 \int {u_{1}}^{\frac{2 \cdot 3}{2 (1)}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.2

Cancel the common factor.

$24 \int {u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.3

Rewrite the expression.

$24 \int {u_{1}}^{\frac{3}{1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.4

Divide $3$ by $1$ .

$24 \int {u_{1}}^{3} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 3.2

Rewrite ${\sqrt{u_{1}}}^{8}$ as ${u_{1}}^{4}$ .

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Step 3.2.1

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

$24 \int {u_{1}}^{3} e^{- {({u_{1}}^{\frac{1}{2}})}^{8}} \frac{1}{2} d u_{1}$

Step 3.2.2

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

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{1}{2} \cdot 8}} \frac{1}{2} d u_{1}$

Step 3.2.3

Combine $\frac{1}{2}$ and $8$ .

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{8}{2}}} \frac{1}{2} d u_{1}$

Step 3.2.4

Cancel the common factor of $8$ and $2$ .

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Step 3.2.4.1

Factor $2$ out of $8$ .

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $2$ .

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.2

Cancel the common factor.

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.3

Rewrite the expression.

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{\frac{4}{1}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.4

Divide $4$ by $1$ .

$24 \int {u_{1}}^{3} e^{- {u_{1}}^{4}} \frac{1}{2} d u_{1}$

Step 3.3

Combine $\frac{1}{2}$ and ${u_{1}}^{3}$ .

$24 \int \frac{{u_{1}}^{3}}{2} e^{- {u_{1}}^{4}} d u_{1}$

Step 3.4

Combine $\frac{{u_{1}}^{3}}{2}$ and $e^{- {u_{1}}^{4}}$ .

$24 \int \frac{{u_{1}}^{3} e^{- {u_{1}}^{4}}}{2} d u_{1}$

Step 4

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

$24 (\frac{1}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $24$ .

$\frac{24}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 5.2

Cancel the common factor of $24$ and $2$ .

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Step 5.2.1

Factor $2$ out of $24$ .

$\frac{2 \cdot 12}{2} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 12}{2 (1)} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 12}{2 \cdot 1} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 5.2.2.3

Rewrite the expression.

$\frac{12}{1} \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 5.2.2.4

Divide $12$ by $1$ .

$12 \int {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

Step 6

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 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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 6.2

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

$12 \int u_{2} e^{- {\sqrt{u_{2}}}^{4}} \frac{1}{2} d u_{2}$

Step 7

Simplify.

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Step 7.1

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

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Step 7.1.1

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

$12 \int u_{2} e^{- {({u_{2}}^{\frac{1}{2}})}^{4}} \frac{1}{2} d u_{2}$

Step 7.1.2

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

$12 \int u_{2} e^{- {u_{2}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{2}$

Step 7.1.3

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

$12 \int u_{2} e^{- {u_{2}}^{\frac{4}{2}}} \frac{1}{2} d u_{2}$

Step 7.1.4

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

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Step 7.1.4.1

Factor $2$ out of $4$ .

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

Step 7.1.4.2

Cancel the common factors.

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Step 7.1.4.2.1

Factor $2$ out of $2$ .

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

Step 7.1.4.2.2

Cancel the common factor.

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

Step 7.1.4.2.3

Rewrite the expression.

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

Step 7.1.4.2.4

Divide $2$ by $1$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Factor $2$ out of $12$ .

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

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $2$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

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

Step 9.2.2.4

Divide $6$ by $1$ .

$6 \int u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 10

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

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

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

$- (2 u_{2})$

Step 10.1.4

Multiply $2$ by $- 1$ .

$- 2 u_{2}$

Step 10.2

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

$6 \int e^{u_{3}} \frac{1}{- 2} d u_{3}$

Step 11

Simplify.

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Step 11.1

Move the negative in front of the fraction.

$6 \int e^{u_{3}} (- \frac{1}{2}) d u_{3}$

Step 11.2

Combine $e^{u_{3}}$ and $\frac{1}{2}$ .

$6 \int - \frac{e^{u_{3}}}{2} d u_{3}$

Step 12

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

$6 (- \int \frac{e^{u_{3}}}{2} d u_{3})$

Step 13

Multiply $- 1$ by $6$ .

$- 6 \int \frac{e^{u_{3}}}{2} d u_{3}$

Step 14

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$- 6 (\frac{1}{2} \int e^{u_{3}} d u_{3})$

Step 15

Simplify.

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Step 15.1

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

$\frac{- 6}{2} \int e^{u_{3}} d u_{3}$

Step 15.2

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

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Step 15.2.1

Factor $2$ out of $- 6$ .

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

Step 15.2.2

Cancel the common factors.

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Step 15.2.2.1

Factor $2$ out of $2$ .

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

Step 15.2.2.2

Cancel the common factor.

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

Step 15.2.2.3

Rewrite the expression.

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

Step 15.2.2.4

Divide $- 3$ by $1$ .

$- 3 \int e^{u_{3}} d u_{3}$

Step 16

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

$- 3 (e^{u_{3}} + C)$

Step 17

Simplify.

$- 3 e^{u_{3}} + C$

Step 18

Substitute back in for each integration substitution variable.

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Step 18.1

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

$- 3 e^{- {u_{2}}^{2}} + C$

Step 18.2

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

$- 3 e^{- {({u_{1}}^{2})}^{2}} + C$

Step 18.3

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

$- 3 e^{- {({(t^{2})}^{2})}^{2}} + C$