Calculus Examples

${(2 + e^{3 x})}^{2}$

Step 1

Write ${(2 + e^{3 x})}^{2}$ as a function.

$f (x) = {(2 + e^{3 x})}^{2}$

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.

$F (x) = \int {(2 + e^{3 x})}^{2} d x$

Step 4

Simplify.

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Step 4.1

Rewrite ${(2 + e^{3 x})}^{2}$ as $(2 + e^{3 x}) (2 + e^{3 x})$ .

$\int (2 + e^{3 x}) (2 + e^{3 x}) d x$

Step 4.2

Expand $(2 + e^{3 x}) (2 + e^{3 x})$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\int 2 (2 + e^{3 x}) + e^{3 x} (2 + e^{3 x}) d x$

Step 4.2.2

Apply the distributive property.

$\int 2 \cdot 2 + 2 e^{3 x} + e^{3 x} (2 + e^{3 x}) d x$

Step 4.2.3

Apply the distributive property.

$\int 2 \cdot 2 + 2 e^{3 x} + e^{3 x} \cdot 2 + e^{3 x} e^{3 x} d x$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $2$ by $2$ .

$\int 4 + 2 e^{3 x} + e^{3 x} \cdot 2 + e^{3 x} e^{3 x} d x$

Step 4.3.1.2

Move $2$ to the left of $e^{3 x}$ .

$\int 4 + 2 e^{3 x} + 2 \cdot e^{3 x} + e^{3 x} e^{3 x} d x$

Step 4.3.1.3

Multiply $e^{3 x}$ by $e^{3 x}$ by adding the exponents.

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Step 4.3.1.3.1

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

$\int 4 + 2 e^{3 x} + 2 e^{3 x} + e^{3 x + 3 x} d x$

Step 4.3.1.3.2

Add $3 x$ and $3 x$ .

$\int 4 + 2 e^{3 x} + 2 e^{3 x} + e^{6 x} d x$

Step 4.3.2

Add $2 e^{3 x}$ and $2 e^{3 x}$ .

$\int 4 + 4 e^{3 x} + e^{6 x} d x$

Step 5

Split the single integral into multiple integrals.

$\int 4 d x + \int 4 e^{3 x} d x + \int e^{6 x} d x$

Step 6

Apply the constant rule.

$4 x + C + \int 4 e^{3 x} d x + \int e^{6 x} d x$

Step 7

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 x + C + 4 \int e^{3 x} d x + \int e^{6 x} d x$

Step 8

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 8.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 8.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 8.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 8.1.3

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

$3 \cdot 1$

Step 8.1.4

Multiply $3$ by $1$ .

$3$

Step 8.2

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

$4 x + C + 4 \int e^{u_{1}} \frac{1}{3} d u_{1} + \int e^{6 x} d x$

Step 9

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

$4 x + C + 4 \int \frac{e^{u_{1}}}{3} d u_{1} + \int e^{6 x} d x$

Step 10

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$4 x + C + 4 (\frac{1}{3} \int e^{u_{1}} d u_{1}) + \int e^{6 x} d x$

Step 11

Combine $\frac{1}{3}$ and $4$ .

$4 x + C + \frac{4}{3} \int e^{u_{1}} d u_{1} + \int e^{6 x} d x$

Step 12

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

$4 x + C + \frac{4}{3} (e^{u_{1}} + C) + \int e^{6 x} d x$

Step 13

Let $u_{2} = 6 x$ . Then $d u_{2} = 6 d x$ , so $\frac{1}{6} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 13.1

Let $u_{2} = 6 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 13.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 13.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 13.1.3

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

$6 \cdot 1$

Step 13.1.4

Multiply $6$ by $1$ .

$6$

Step 13.2

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

$4 x + C + \frac{4}{3} (e^{u_{1}} + C) + \int e^{u_{2}} \frac{1}{6} d u_{2}$

Step 14

Combine $e^{u_{2}}$ and $\frac{1}{6}$ .

$4 x + C + \frac{4}{3} (e^{u_{1}} + C) + \int \frac{e^{u_{2}}}{6} d u_{2}$

Step 15

Since $\frac{1}{6}$ is constant with respect to $u_{2}$ , move $\frac{1}{6}$ out of the integral.

$4 x + C + \frac{4}{3} (e^{u_{1}} + C) + \frac{1}{6} \int e^{u_{2}} d u_{2}$

Step 16

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

$4 x + C + \frac{4}{3} (e^{u_{1}} + C) + \frac{1}{6} (e^{u_{2}} + C)$

Step 17

Simplify.

$4 x + \frac{4}{3} e^{u_{1}} + \frac{1}{6} e^{u_{2}} + C$

Step 18

Substitute back in for each integration substitution variable.

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Step 18.1

Replace all occurrences of $u_{1}$ with $3 x$ .

$4 x + \frac{4}{3} e^{3 x} + \frac{1}{6} e^{u_{2}} + C$

Step 18.2

Replace all occurrences of $u_{2}$ with $6 x$ .

$4 x + \frac{4}{3} e^{3 x} + \frac{1}{6} e^{6 x} + C$

Step 19

The answer is the antiderivative of the function $f (x) = {(2 + e^{3 x})}^{2}$ .

$F (x) =$ $4 x + \frac{4}{3} e^{3 x} + \frac{1}{6} e^{6 x} + C$