Calculus Examples

$\int 2 e^{2 y} - 2 e^{- 2 y} d y$

Step 1

Split the single integral into multiple integrals.

$\int 2 e^{2 y} d y + \int - 2 e^{- 2 y} d y$

Step 2

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int e^{2 y} d y + \int - 2 e^{- 2 y} d y$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $2 y$ .

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

Step 3.1.2

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 3.1.3

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 3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

$1 \int e^{u_{1}} d u_{1} + \int - 2 e^{- 2 y} d y$

Step 6.3

Multiply $\int e^{u_{1}} d u_{1}$ by $1$ .

$\int e^{u_{1}} d u_{1} + \int - 2 e^{- 2 y} d y$

Step 7

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$e^{u_{1}} + C + \int - 2 e^{- 2 y} d y$

Step 8

Since $- 2$ is constant with respect to $y$ , move $- 2$ out of the integral.

$e^{u_{1}} + C - 2 \int e^{- 2 y} d y$

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $- 2 y$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 9.2

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

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

Step 10

Simplify.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 11

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

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

Step 12

Multiply $- 1$ by $- 2$ .

$e^{u_{1}} + C + 2 \int \frac{e^{u_{2}}}{2} d u_{2}$

Step 13

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

$e^{u_{1}} + C + 2 (\frac{1}{2} \int e^{u_{2}} d u_{2})$

Step 14

Simplify.

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Step 14.1

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

$e^{u_{1}} + C + \frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 14.2

Cancel the common factor of $2$ .

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Step 14.2.1

Cancel the common factor.

$e^{u_{1}} + C + \frac{2}{2} \int e^{u_{2}} d u_{2}$

Step 14.2.2

Rewrite the expression.

$e^{u_{1}} + C + 1 \int e^{u_{2}} d u_{2}$

Step 14.3

Multiply $\int e^{u_{2}} d u_{2}$ by $1$ .

$e^{u_{1}} + C + \int e^{u_{2}} d u_{2}$

Step 15

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

$e^{u_{1}} + C + e^{u_{2}} + C$

Step 16

Simplify.

$e^{u_{1}} + e^{u_{2}} + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

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

$e^{2 y} + e^{u_{2}} + C$

Step 17.2

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

$e^{2 y} + e^{- 2 y} + C$