Calculus Examples

$\int x - 2 e^{- x} - \frac{1}{2} \cdot e^{- 2 x} d x$

Step 1

Split the single integral into multiple integrals.

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

Step 2

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 3

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

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

Step 4

Let $u_{1} = - x$ . Then $d u_{1} = - d x$ , so $- d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.1

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

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Step 4.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 4.1.2

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

$- \frac{d}{d x} [x]$

Step 4.1.3

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

$- 1 \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2

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

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

Step 5

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

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

Step 6

Multiply $- 1$ by $- 2$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $- 2 x$ .

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

Step 9.1.2

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

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

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ 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 $d u_{2}$ .

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

Step 10

Simplify.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Multiply $- 1$ by $- 1$ .

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

Step 12.2

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

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Multiply $2$ by $2$ .

$\frac{1}{2} x^{2} + C + 2 (e^{u_{1}} + C) + \frac{1}{4} \int e^{u_{2}} d u_{2}$

Step 15

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

$\frac{1}{2} x^{2} + C + 2 (e^{u_{1}} + C) + \frac{1}{4} (e^{u_{2}} + C)$

Step 16

Simplify.

$\frac{1}{2} x^{2} + 2 e^{u_{1}} + \frac{1}{4} e^{u_{2}} + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

Replace all occurrences of $u_{1}$ with $- x$ .

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

Step 17.2

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

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