Calculus Examples

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

Step 1

Simplify the expression.

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Step 1.1

Negate the exponent of $e^{2 y}$ and move it out of the denominator.

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $- 1$ by $2$ .

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

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{- 2 y}$ .

$y (- \frac{1}{2} e^{- 2 y})]_{0}^{1} - \int_{0}^{1} - \frac{1}{2} e^{- 2 y} d y$

Step 3

Simplify.

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Step 3.1

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

$y (- \frac{e^{- 2 y}}{2})]_{0}^{1} - \int_{0}^{1} - \frac{1}{2} e^{- 2 y} d y$

Step 3.2

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} - \int_{0}^{1} - \frac{1}{2} e^{- 2 y} d y$

Step 4

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} - (- \frac{1}{2} \int_{0}^{1} e^{- 2 y} d y)$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + 1 (\frac{1}{2} \int_{0}^{1} e^{- 2 y} d y)$

Step 5.2

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} \int_{0}^{1} e^{- 2 y} d y$

Step 6

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

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Step 6.1

Let $u = - 2 y$ . Find $\frac{d u}{d y}$ .

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Step 6.1.1

Differentiate $- 2 y$ .

$\frac{d}{d y} [- 2 y]$

Step 6.1.2

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

$- 2 \frac{d}{d y} [y]$

Step 6.1.3

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

$- 2 \cdot 1$

Step 6.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.2

Substitute the lower limit in for $y$ in $u = - 2 y$ .

$u_{lower} = - 2 \cdot 0$

Step 6.3

Multiply $- 2$ by $0$ .

$u_{lower} = 0$

Step 6.4

Substitute the upper limit in for $y$ in $u = - 2 y$ .

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

Step 6.5

Multiply $- 2$ by $1$ .

$u_{upper} = - 2$

Step 6.6

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

$u_{lower} = 0$

$u_{upper} = - 2$

Step 6.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} \int_{0}^{- 2} e^{u} \frac{1}{- 2} d u$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} \int_{0}^{- 2} e^{u} (- \frac{1}{2}) d u$

Step 7.2

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} \int_{0}^{- 2} - \frac{e^{u}}{2} d u$

Step 8

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} (- \int_{0}^{- 2} \frac{e^{u}}{2} d u)$

Step 9

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} + \frac{1}{2} (- (\frac{1}{2} \int_{0}^{- 2} e^{u} d u))$

Step 10

Simplify.

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Step 10.1

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} - \frac{1}{2 \cdot 2} \int_{0}^{- 2} e^{u} d u$

Step 10.2

Multiply $2$ by $2$ .

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} - \frac{1}{4} \int_{0}^{- 2} e^{u} d u$

Step 11

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

$- \frac{y e^{- 2 y}}{2}]_{0}^{1} - \frac{1}{4} e^{u}]_{0}^{- 2}$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $- \frac{y e^{- 2 y}}{2}$ at $1$ and at $0$ .

$(- \frac{1 e^{- 2 \cdot 1}}{2}) + \frac{0 e^{- 2 \cdot 0}}{2} - \frac{1}{4} e^{u}]_{0}^{- 2}$

Step 12.2

Evaluate $e^{u}$ at $- 2$ and at $0$ .

$(- \frac{1 e^{- 2 \cdot 1}}{2}) + \frac{0 e^{- 2 \cdot 0}}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3

Simplify.

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Step 12.3.1

Multiply $- 2$ by $1$ .

$- \frac{1 e^{- 2}}{2} + \frac{0 e^{- 2 \cdot 0}}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.2

Multiply $e^{- 2}$ by $1$ .

$- \frac{e^{- 2}}{2} + \frac{0 e^{- 2 \cdot 0}}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.3

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

$- \frac{1}{2 e^{2}} + \frac{0 e^{- 2 \cdot 0}}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.4

Multiply $- 2$ by $0$ .

$- \frac{1}{2 e^{2}} + \frac{0 e^{0}}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.5

Anything raised to $0$ is $1$ .

$- \frac{1}{2 e^{2}} + \frac{0 \cdot 1}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.6

Multiply $0$ by $1$ .

$- \frac{1}{2 e^{2}} + \frac{0}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.7

Cancel the common factor of $0$ and $2$ .

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Step 12.3.7.1

Factor $2$ out of $0$ .

$- \frac{1}{2 e^{2}} + \frac{2 (0)}{2} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.7.2

Cancel the common factors.

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Step 12.3.7.2.1

Factor $2$ out of $2$ .

$- \frac{1}{2 e^{2}} + \frac{2 \cdot 0}{2 \cdot 1} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.7.2.2

Cancel the common factor.

$- \frac{1}{2 e^{2}} + \frac{2 \cdot 0}{2 \cdot 1} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.7.2.3

Rewrite the expression.

$- \frac{1}{2 e^{2}} + \frac{0}{1} - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.7.2.4

Divide $0$ by $1$ .

$- \frac{1}{2 e^{2}} + 0 - \frac{1}{4} ((e^{- 2}) - e^{0})$

Step 12.3.8

Add $- \frac{1}{2 e^{2}}$ and $0$ .

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

Step 12.3.9

Anything raised to $0$ is $1$ .

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

Step 12.3.10

Multiply $- 1$ by $1$ .

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

Step 12.3.11

To write $- \frac{1}{4} (e^{- 2} - 1)$ as a fraction with a common denominator, multiply by $\frac{2 e^{2}}{2 e^{2}}$ .

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

Step 12.3.12

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

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

Step 12.3.13

Combine the numerators over the common denominator.

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

Step 12.3.14

Multiply $2$ by $- 1$ .

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

Step 12.3.15

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

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

Step 12.3.16

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

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

Step 12.3.17

Move $- 2$ to the left of $e^{2}$ .

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

Step 12.3.18

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

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Step 12.3.18.1

Factor $2$ out of $- 2 e^{2}$ .

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

Step 12.3.18.2

Cancel the common factors.

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Step 12.3.18.2.1

Factor $2$ out of $4$ .

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

Step 12.3.18.2.2

Cancel the common factor.

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

Step 12.3.18.2.3

Rewrite the expression.

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

Step 12.3.19

Move the negative in front of the fraction.

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

Step 13

Simplify.

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Step 13.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 13.2

Factor $- 1$ out of $- \frac{e^{2}}{2} (e^{- 2} - 1)$ .

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

Step 13.3

Factor $- 1$ out of $- 1 (1) - (\frac{e^{2}}{2} (e^{- 2} - 1))$ .

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

Step 13.4

Move the negative in front of the fraction.

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

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Multiply $\frac{e^{2}}{2} e^{- 2}$ .

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Step 14.2.1

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

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

Step 14.2.2

Multiply $e^{2}$ by $e^{- 2}$ by adding the exponents.

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Step 14.2.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 14.2.2.2

Subtract $2$ from $2$ .

$- \frac{1 + \frac{e^{0}}{2} + \frac{e^{2}}{2} \cdot - 1}{2 e^{2}}$

Step 14.2.3

Simplify $e^{0}$ .

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

Step 14.3

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

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

Step 14.4

Move $- 1$ to the left of $e^{2}$ .

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

Step 14.5

Rewrite $- 1 e^{2}$ as $- e^{2}$ .

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

Step 14.6

Simplify the numerator.

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Step 14.6.1

Write $1$ as a fraction with a common denominator.

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

Step 14.6.2

Combine the numerators over the common denominator.

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

Step 14.6.3

Add $2$ and $1$ .

$- \frac{\frac{3}{2} + \frac{- e^{2}}{2}}{2 e^{2}}$

Step 14.6.4

Combine the numerators over the common denominator.

$- \frac{\frac{3 - e^{2}}{2}}{2 e^{2}}$

Step 14.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 14.8

Multiply $\frac{3 - e^{2}}{2} \cdot \frac{1}{2 e^{2}}$ .

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Step 14.8.1

Multiply $\frac{3 - e^{2}}{2}$ by $\frac{1}{2 e^{2}}$ .

$- \frac{3 - e^{2}}{2 (2 e^{2})}$

Step 14.8.2

Multiply $2$ by $2$ .

$- \frac{3 - e^{2}}{4 e^{2}}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 - e^{2}}{4 e^{2}}$

Decimal Form:

$0.14849853 \dots$

Step 16