Calculus Examples

$\int_{0}^{1} x^{7} e^{- x^{8}} d x$

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

Substitute the lower limit in for $x$ in $u_{1} = x^{2}$ .

$u_{lower} = 0^{2}$

Step 1.3

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = x^{2}$ .

$u_{upper} = 1^{2}$

Step 1.5

One to any power is one.

$u_{upper} = 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{0}^{1} {\sqrt{u_{1}}}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

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

$\int_{0}^{1} {({u_{1}}^{\frac{1}{2}})}^{6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.2

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

$\int_{0}^{1} {u_{1}}^{\frac{1}{2} \cdot 6} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.3

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

$\int_{0}^{1} {u_{1}}^{\frac{6}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.4

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

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Step 2.1.4.1

Factor $2$ out of $6$ .

$\int_{0}^{1} {u_{1}}^{\frac{2 \cdot 3}{2}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int_{0}^{1} {u_{1}}^{\frac{2 \cdot 3}{2 (1)}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.2

Cancel the common factor.

$\int_{0}^{1} {u_{1}}^{\frac{2 \cdot 3}{2 \cdot 1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.3

Rewrite the expression.

$\int_{0}^{1} {u_{1}}^{\frac{3}{1}} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.1.4.2.4

Divide $3$ by $1$ .

$\int_{0}^{1} {u_{1}}^{3} e^{- {\sqrt{u_{1}}}^{8}} \frac{1}{2} d u_{1}$

Step 2.2

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

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Step 2.2.1

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

$\int_{0}^{1} {u_{1}}^{3} e^{- {({u_{1}}^{\frac{1}{2}})}^{8}} \frac{1}{2} d u_{1}$

Step 2.2.2

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

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{1}{2} \cdot 8}} \frac{1}{2} d u_{1}$

Step 2.2.3

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

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{8}{2}}} \frac{1}{2} d u_{1}$

Step 2.2.4

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

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Step 2.2.4.1

Factor $2$ out of $8$ .

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $2$ out of $2$ .

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 (1)}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.2

Cancel the common factor.

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{2 \cdot 4}{2 \cdot 1}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.3

Rewrite the expression.

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{\frac{4}{1}}} \frac{1}{2} d u_{1}$

Step 2.2.4.2.4

Divide $4$ by $1$ .

$\int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{4}} \frac{1}{2} d u_{1}$

Step 2.3

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

$\int_{0}^{1} \frac{{u_{1}}^{3}}{2} e^{- {u_{1}}^{4}} d u_{1}$

Step 2.4

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

$\int_{0}^{1} \frac{{u_{1}}^{3} e^{- {u_{1}}^{4}}}{2} d u_{1}$

Step 3

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

$\frac{1}{2} \int_{0}^{1} {u_{1}}^{3} e^{- {u_{1}}^{4}} d u_{1}$

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

Substitute the lower limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

$u_{lower} = 0^{2}$

Step 4.3

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = {u_{1}}^{2}$ .

$u_{upper} = 1^{2}$

Step 4.5

One to any power is one.

$u_{upper} = 1$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {\sqrt{u_{2}}}^{4}} \frac{1}{2} d u_{2}$

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

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {({u_{2}}^{\frac{1}{2}})}^{4}} \frac{1}{2} d u_{2}$

Step 5.1.2

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

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{1}{2} \cdot 4}} \frac{1}{2} d u_{2}$

Step 5.1.3

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

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{4}{2}}} \frac{1}{2} d u_{2}$

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 $4$ .

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2}}} \frac{1}{2} d u_{2}$

Step 5.1.4.2

Cancel the common factors.

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Step 5.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2 (1)}}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.2

Cancel the common factor.

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{2 \cdot 2}{2 \cdot 1}}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.3

Rewrite the expression.

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{\frac{2}{1}}} \frac{1}{2} d u_{2}$

Step 5.1.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{2}} \frac{1}{2} d u_{2}$

Step 5.2

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

$\frac{1}{2} \int_{0}^{1} \frac{u_{2}}{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 5.3

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

$\frac{1}{2} \int_{0}^{1} \frac{u_{2} e^{- {u_{2}}^{2}}}{2} d u_{2}$

Step 6

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

$\frac{1}{2} (\frac{1}{2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{2}} d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{2 \cdot 2} \int_{0}^{1} u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 7.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{0}^{1} u_{2} e^{- {u_{2}}^{2}} d u_{2}$

Step 8

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 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $2$ by $- 1$ .

$- 2 u_{2}$

Step 8.2

Substitute the lower limit in for $u_{2}$ in $u_{3} = - {u_{2}}^{2}$ .

$u_{lower} = - 0^{2}$

Step 8.3

Simplify.

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Step 8.3.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = - 0$

Step 8.3.2

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 8.4

Substitute the upper limit in for $u_{2}$ in $u_{3} = - {u_{2}}^{2}$ .

$u_{upper} = - 1^{2}$

Step 8.5

Simplify.

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Step 8.5.1

One to any power is one.

$u_{upper} = - 1 \cdot 1$

Step 8.5.2

Multiply $- 1$ by $1$ .

$u_{upper} = - 1$

Step 8.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 1$

Step 8.7

Rewrite the problem using $u_{3}$ , $d u_{3}$ , and the new limits of integration.

$\frac{1}{4} \int_{0}^{- 1} e^{u_{3}} \frac{1}{- 2} d u_{3}$

Step 9

Simplify.

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Step 9.1

Move the negative in front of the fraction.

$\frac{1}{4} \int_{0}^{- 1} e^{u_{3}} (- \frac{1}{2}) d u_{3}$

Step 9.2

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

$\frac{1}{4} \int_{0}^{- 1} - \frac{e^{u_{3}}}{2} d u_{3}$

Step 10

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

$\frac{1}{4} (- \int_{0}^{- 1} \frac{e^{u_{3}}}{2} d u_{3})$

Step 11

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

$\frac{1}{4} (- (\frac{1}{2} \int_{0}^{- 1} e^{u_{3}} d u_{3}))$

Step 12

Simplify.

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Step 12.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$- \frac{1}{4 \cdot 2} \int_{0}^{- 1} e^{u_{3}} d u_{3}$

Step 12.2

Multiply $4$ by $2$ .

$- \frac{1}{8} \int_{0}^{- 1} e^{u_{3}} d u_{3}$

Step 13

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

$- \frac{1}{8} e^{u_{3}}]_{0}^{- 1}$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $e^{u_{3}}$ at $- 1$ and at $0$ .

$- \frac{1}{8} ((e^{- 1}) - e^{0})$

Step 14.2

Simplify.

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Step 14.2.1

Anything raised to $0$ is $1$ .

$- \frac{1}{8} (e^{- 1} - 1 \cdot 1)$

Step 14.2.2

Multiply $- 1$ by $1$ .

$- \frac{1}{8} (e^{- 1} - 1)$

Step 15

Simplify.

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Step 15.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{8} (\frac{1}{e} - 1)$

Step 15.2

Apply the distributive property.

$- \frac{1}{8} \cdot \frac{1}{e} - \frac{1}{8} \cdot - 1$

Step 15.3

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

$- \frac{1}{e \cdot 8} - \frac{1}{8} \cdot - 1$

Step 15.4

Multiply $- \frac{1}{8} \cdot - 1$ .

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Step 15.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{e \cdot 8} + 1 (\frac{1}{8})$

Step 15.4.2

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

$- \frac{1}{e \cdot 8} + \frac{1}{8}$

Step 15.5

Move $8$ to the left of $e$ .

$- \frac{1}{8 e} + \frac{1}{8}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{8 e} + \frac{1}{8}$

Decimal Form:

$0.07901506 \dots$

Step 17