Calculus Examples

$\int 30 t^{5} e^{- t^{6}} d t$

Step 1

Since $30$ is constant with respect to $t$ , move $30$ out of the integral.

$30 \int t^{5} e^{- t^{6}} d t$

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 $d u_{1}$ .

$30 \int {\sqrt{u_{1}}}^{4} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

$30 \int {({u_{1}}^{\frac{1}{2}})}^{4} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.2

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

$30 \int {u_{1}}^{\frac{1}{2} \cdot 4} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.3

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

$30 \int {u_{1}}^{\frac{4}{2}} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.4

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

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Step 3.1.4.1

Factor $2$ out of $4$ .

$30 \int {u_{1}}^{\frac{2 \cdot 2}{2}} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

$30 \int {u_{1}}^{\frac{2 \cdot 2}{2 (1)}} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.2

Cancel the common factor.

$30 \int {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.3

Rewrite the expression.

$30 \int {u_{1}}^{\frac{2}{1}} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.1.4.2.4

Divide $2$ by $1$ .

$30 \int {u_{1}}^{2} e^{- {\sqrt{u_{1}}}^{6}} \frac{1}{2} d u_{1}$

Step 3.2

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

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Step 3.2.1

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

$30 \int {u_{1}}^{2} e^{- {({u_{1}}^{\frac{1}{2}})}^{6}} \frac{1}{2} d u_{1}$

Step 3.2.2

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

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{1}{2} \cdot 6}} \frac{1}{2} d u_{1}$

Step 3.2.3

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

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{6}{2}}} \frac{1}{2} d u_{1}$

Step 3.2.4

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

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Step 3.2.4.1

Factor $2$ out of $6$ .

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{2 \cdot 3}{2}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $2$ .

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{2 \cdot 3}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.2

Cancel the common factor.

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.3

Rewrite the expression.

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{\frac{3}{1}}} \frac{1}{2} d u_{1}$

Step 3.2.4.2.4

Divide $3$ by $1$ .

$30 \int {u_{1}}^{2} e^{- {u_{1}}^{3}} \frac{1}{2} d u_{1}$

Step 3.3

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

$30 \int \frac{{u_{1}}^{2}}{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 3.4

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

$30 \int \frac{{u_{1}}^{2} e^{- {u_{1}}^{3}}}{2} d u_{1}$

Step 4

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

$30 (\frac{1}{2} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $30$ .

$\frac{30}{2} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 5.2

Cancel the common factor of $30$ and $2$ .

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Step 5.2.1

Factor $2$ out of $30$ .

$\frac{2 \cdot 15}{2} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 15}{2 (1)} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 15}{2 \cdot 1} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 5.2.2.3

Rewrite the expression.

$\frac{15}{1} \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 5.2.2.4

Divide $15$ by $1$ .

$15 \int {u_{1}}^{2} e^{- {u_{1}}^{3}} d u_{1}$

Step 6

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

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Step 6.1

Let $u_{2} = - {u_{1}}^{3}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $- {u_{1}}^{3}$ .

$\frac{d}{d u_{1}} [- {u_{1}}^{3}]$

Step 6.1.2

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- {u_{1}}^{3}$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [{u_{1}}^{3}]$ .

$- \frac{d}{d u_{1}} [{u_{1}}^{3}]$

Step 6.1.3

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

$- (3 {u_{1}}^{2})$

Step 6.1.4

Multiply $3$ by $- 1$ .

$- 3 {u_{1}}^{2}$

Step 6.2

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

$15 \int e^{u_{2}} \frac{1}{- 3} d u_{2}$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$15 \int e^{u_{2}} (- \frac{1}{3}) d u_{2}$

Step 7.2

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

$15 \int - \frac{e^{u_{2}}}{3} d u_{2}$

Step 8

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

$15 (- \int \frac{e^{u_{2}}}{3} d u_{2})$

Step 9

Multiply $- 1$ by $15$ .

$- 15 \int \frac{e^{u_{2}}}{3} d u_{2}$

Step 10

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

$- 15 (\frac{1}{3} \int e^{u_{2}} d u_{2})$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{3}$ and $- 15$ .

$\frac{- 15}{3} \int e^{u_{2}} d u_{2}$

Step 11.2

Cancel the common factor of $- 15$ and $3$ .

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Step 11.2.1

Factor $3$ out of $- 15$ .

$\frac{3 \cdot - 5}{3} \int e^{u_{2}} d u_{2}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 5}{3 (1)} \int e^{u_{2}} d u_{2}$

Step 11.2.2.2

Cancel the common factor.

$\frac{3 \cdot - 5}{3 \cdot 1} \int e^{u_{2}} d u_{2}$

Step 11.2.2.3

Rewrite the expression.

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

Step 11.2.2.4

Divide $- 5$ by $1$ .

$- 5 \int e^{u_{2}} d u_{2}$

Step 12

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

$- 5 (e^{u_{2}} + C)$

Step 13

Simplify.

$- 5 e^{u_{2}} + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{2}$ with $- {u_{1}}^{3}$ .

$- 5 e^{- {u_{1}}^{3}} + C$

Step 14.2

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

$- 5 e^{- {(t^{2})}^{3}} + C$