Calculus Examples

$\int_{1}^{2} \frac{e^{2 x^{- 2}}}{x^{3}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{2} e^{2 x^{- 2}} {(x^{3})}^{- 1} d x$

Step 1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.1

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

$\int_{1}^{2} e^{2 x^{- 2}} x^{3 \cdot - 1} d x$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int_{1}^{2} e^{2 x^{- 2}} x^{- 3} d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

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 2.2

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

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

Step 2.3

One to any power is one.

$u_{lower} = 1$

Step 2.4

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

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

Step 2.5

Raise $2$ to the power of $2$ .

$u_{upper} = 4$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 4$

Step 2.7

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

Factor $2$ out of $- 2$ .

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

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $2$ .

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

Step 3.1.4.2.2

Cancel the common factor.

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

Step 3.1.4.2.3

Rewrite the expression.

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

Step 3.1.4.2.4

Divide $- 1$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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Step 3.2.4.1

Factor $2$ out of $- 4$ .

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

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $2$ .

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

Step 3.2.4.2.2

Cancel the common factor.

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

Step 3.2.4.2.3

Rewrite the expression.

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

Step 3.2.4.2.4

Divide $- 2$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Move ${u_{1}}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

Move ${u_{1}}^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2

Multiply the exponents in ${({u_{1}}^{2})}^{- 1}$ .

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

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 $n = - 1$ .

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

Step 6.1.4

Multiply $- 1$ by $2$ .

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

Step 6.1.5

Simplify.

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Step 6.1.5.1

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

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

Step 6.1.5.2

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

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

Step 6.1.5.3

Move the negative in front of the fraction.

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

Step 6.2

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

$u_{lower} = 2 \cdot 1^{- 1}$

Step 6.3

Simplify.

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Step 6.3.1

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

$u_{lower} = 2 \cdot \frac{1}{1}$

Step 6.3.2

Cancel the common factor of $1$ .

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Step 6.3.2.1

Cancel the common factor.

$u_{lower} = 2 \cdot \frac{1}{1}$

Step 6.3.2.2

Rewrite the expression.

$u_{lower} = 2 \cdot 1$

Step 6.3.3

Multiply $2$ by $1$ .

$u_{lower} = 2$

Step 6.4

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

$u_{upper} = 2 \cdot 4^{- 1}$

Step 6.5

Simplify.

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Step 6.5.1

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

$u_{upper} = 2 \cdot \frac{1}{4}$

Step 6.5.2

Cancel the common factor of $2$ .

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Step 6.5.2.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \cdot \frac{1}{2 (2)}$

Step 6.5.2.2

Cancel the common factor.

$u_{upper} = 2 \cdot \frac{1}{2 \cdot 2}$

Step 6.5.2.3

Rewrite the expression.

$u_{upper} = \frac{1}{2}$

Step 6.6

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

$u_{lower} = 2$

$u_{upper} = \frac{1}{2}$

Step 6.7

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 9

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

$- \frac{1}{4} e^{u_{2}}]_{2}^{\frac{1}{2}}$

Step 10

Evaluate $e^{u_{2}}$ at $\frac{1}{2}$ and at $2$ .

$- \frac{1}{4} (e^{\frac{1}{2}} - e^{2})$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$- \frac{1}{4} e^{\frac{1}{2}} - \frac{1}{4} (- e^{2})$

Step 11.2

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

$- \frac{e^{\frac{1}{2}}}{4} - \frac{1}{4} (- e^{2})$

Step 11.3

Multiply $- \frac{1}{4} (- e^{2})$ .

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Step 11.3.1

Multiply $- 1$ by $- 1$ .

$- \frac{e^{\frac{1}{2}}}{4} + 1 (\frac{1}{4}) e^{2}$

Step 11.3.2

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

$- \frac{e^{\frac{1}{2}}}{4} + \frac{1}{4} e^{2}$

Step 11.3.3

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

$- \frac{e^{\frac{1}{2}}}{4} + \frac{e^{2}}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{e^{\frac{1}{2}}}{4} + \frac{e^{2}}{4}$

Decimal Form:

$1.43508370 \dots$

Step 13