Calculus Examples

$36 e^{- 4 x} + 24 e^{- 2 x} + 4$

Step 1

Write $36 e^{- 4 x} + 24 e^{- 2 x} + 4$ as a function.

$f (x) = 36 e^{- 4 x} + 24 e^{- 2 x} + 4$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 36 e^{- 4 x} + 24 e^{- 2 x} + 4 d x$

Step 4

Split the single integral into multiple integrals.

$\int 36 e^{- 4 x} d x + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 5

Since $36$ is constant with respect to $x$ , move $36$ out of the integral.

$36 \int e^{- 4 x} d x + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 6

Let $u_{1} = - 4 x$ . Then $d u_{1} = - 4 d x$ , so $- \frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 6.1

Let $u_{1} = - 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 6.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 6.1.3

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

$- 4 \cdot 1$

Step 6.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 6.2

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

$36 \int e^{u_{1}} \frac{1}{- 4} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$36 \int e^{u_{1}} (- \frac{1}{4}) d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 7.2

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

$36 \int - \frac{e^{u_{1}}}{4} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 8

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

$36 (- \int \frac{e^{u_{1}}}{4} d u_{1}) + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 9

Multiply $- 1$ by $36$ .

$- 36 \int \frac{e^{u_{1}}}{4} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 10

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$- 36 (\frac{1}{4} \int e^{u_{1}} d u_{1}) + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{4}$ and $- 36$ .

$\frac{- 36}{4} \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11.2

Cancel the common factor of $- 36$ and $4$ .

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Step 11.2.1

Factor $4$ out of $- 36$ .

$\frac{4 \cdot - 9}{4} \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 9}{4 (1)} \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11.2.2.2

Cancel the common factor.

$\frac{4 \cdot - 9}{4 \cdot 1} \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11.2.2.3

Rewrite the expression.

$\frac{- 9}{1} \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 11.2.2.4

Divide $- 9$ by $1$ .

$- 9 \int e^{u_{1}} d u_{1} + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 12

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

$- 9 (e^{u_{1}} + C) + \int 24 e^{- 2 x} d x + \int 4 d x$

Step 13

Since $24$ is constant with respect to $x$ , move $24$ out of the integral.

$- 9 (e^{u_{1}} + C) + 24 \int e^{- 2 x} d x + \int 4 d x$

Step 14

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 14.1

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

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Step 14.1.1

Differentiate $- 2 x$ .

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

Step 14.1.2

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 14.1.3

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 14.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 14.2

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

$- 9 (e^{u_{1}} + C) + 24 \int e^{u_{2}} \frac{1}{- 2} d u_{2} + \int 4 d x$

Step 15

Simplify.

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Step 15.1

Move the negative in front of the fraction.

$- 9 (e^{u_{1}} + C) + 24 \int e^{u_{2}} (- \frac{1}{2}) d u_{2} + \int 4 d x$

Step 15.2

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

$- 9 (e^{u_{1}} + C) + 24 \int - \frac{e^{u_{2}}}{2} d u_{2} + \int 4 d x$

Step 16

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

$- 9 (e^{u_{1}} + C) + 24 (- \int \frac{e^{u_{2}}}{2} d u_{2}) + \int 4 d x$

Step 17

Multiply $- 1$ by $24$ .

$- 9 (e^{u_{1}} + C) - 24 \int \frac{e^{u_{2}}}{2} d u_{2} + \int 4 d x$

Step 18

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

$- 9 (e^{u_{1}} + C) - 24 (\frac{1}{2} \int e^{u_{2}} d u_{2}) + \int 4 d x$

Step 19

Simplify.

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Step 19.1

Combine $\frac{1}{2}$ and $- 24$ .

$- 9 (e^{u_{1}} + C) + \frac{- 24}{2} \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 19.2

Cancel the common factor of $- 24$ and $2$ .

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Step 19.2.1

Factor $2$ out of $- 24$ .

$- 9 (e^{u_{1}} + C) + \frac{2 \cdot - 12}{2} \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 19.2.2

Cancel the common factors.

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Step 19.2.2.1

Factor $2$ out of $2$ .

$- 9 (e^{u_{1}} + C) + \frac{2 \cdot - 12}{2 (1)} \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 19.2.2.2

Cancel the common factor.

$- 9 (e^{u_{1}} + C) + \frac{2 \cdot - 12}{2 \cdot 1} \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 19.2.2.3

Rewrite the expression.

$- 9 (e^{u_{1}} + C) + \frac{- 12}{1} \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 19.2.2.4

Divide $- 12$ by $1$ .

$- 9 (e^{u_{1}} + C) - 12 \int e^{u_{2}} d u_{2} + \int 4 d x$

Step 20

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

$- 9 (e^{u_{1}} + C) - 12 (e^{u_{2}} + C) + \int 4 d x$

Step 21

Apply the constant rule.

$- 9 (e^{u_{1}} + C) - 12 (e^{u_{2}} + C) + 4 x + C$

Step 22

Simplify.

$- 9 e^{u_{1}} - 12 e^{u_{2}} + 4 x + C$

Step 23

Substitute back in for each integration substitution variable.

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Step 23.1

Replace all occurrences of $u_{1}$ with $- 4 x$ .

$- 9 e^{- 4 x} - 12 e^{u_{2}} + 4 x + C$

Step 23.2

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

$- 9 e^{- 4 x} - 12 e^{- 2 x} + 4 x + C$

Step 24

The answer is the antiderivative of the function $f (x) = 36 e^{- 4 x} + 24 e^{- 2 x} + 4$ .

$F (x) =$ $- 9 e^{- 4 x} - 12 e^{- 2 x} + 4 x + C$