Calculus Examples

$f (x) = e^{7 x} + 5 \sec^{2} (3 x)$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int e^{7 x} + 5 \sec^{2} (3 x) d x$

Step 3

Split the single integral into multiple integrals.

$\int e^{7 x} d x + \int 5 \sec^{2} (3 x) d x$

Step 4

Let $u_{1} = 7 x$ . Then $d u_{1} = 7 d x$ , so $\frac{1}{7} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.1

Let $u_{1} = 7 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 4.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 4.1.3

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

$7 \cdot 1$

Step 4.1.4

Multiply $7$ by $1$ .

$7$

Step 4.2

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

$\int e^{u_{1}} \frac{1}{7} d u_{1} + \int 5 \sec^{2} (3 x) d x$

Step 5

Combine $e^{u_{1}}$ and $\frac{1}{7}$ .

$\int \frac{e^{u_{1}}}{7} d u_{1} + \int 5 \sec^{2} (3 x) d x$

Step 6

Since $\frac{1}{7}$ is constant with respect to $u_{1}$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int e^{u_{1}} d u_{1} + \int 5 \sec^{2} (3 x) d x$

Step 7

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

$\frac{1}{7} (e^{u_{1}} + C) + \int 5 \sec^{2} (3 x) d x$

Step 8

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{7} (e^{u_{1}} + C) + 5 \int \sec^{2} (3 x) d x$

Step 9

Let $u_{2} = 3 x$ . Then $d u_{2} = 3 d x$ , so $\frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 9.1

Let $u_{2} = 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 9.1.1

Differentiate $3 x$ .

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

Step 9.1.2

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 9.1.3

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

$3 \cdot 1$

Step 9.1.4

Multiply $3$ by $1$ .

$3$

Step 9.2

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

$\frac{1}{7} (e^{u_{1}} + C) + 5 \int \sec^{2} (u_{2}) \frac{1}{3} d u_{2}$

Step 10

Combine $\sec^{2} (u_{2})$ and $\frac{1}{3}$ .

$\frac{1}{7} (e^{u_{1}} + C) + 5 \int \frac{\sec^{2} (u_{2})}{3} d u_{2}$

Step 11

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{7} (e^{u_{1}} + C) + 5 (\frac{1}{3} \int \sec^{2} (u_{2}) d u_{2})$

Step 12

Combine $\frac{1}{3}$ and $5$ .

$\frac{1}{7} (e^{u_{1}} + C) + \frac{5}{3} \int \sec^{2} (u_{2}) d u_{2}$

Step 13

Since the derivative of $\tan (u_{2})$ is $\sec^{2} (u_{2})$ , the integral of $\sec^{2} (u_{2})$ is $\tan (u_{2})$ .

$\frac{1}{7} (e^{u_{1}} + C) + \frac{5}{3} (\tan (u_{2}) + C)$

Step 14

Simplify.

$\frac{1}{7} e^{u_{1}} + \frac{5}{3} \tan (u_{2}) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{1}$ with $7 x$ .

$\frac{1}{7} e^{7 x} + \frac{5}{3} \tan (u_{2}) + C$

Step 15.2

Replace all occurrences of $u_{2}$ with $3 x$ .

$\frac{1}{7} e^{7 x} + \frac{5}{3} \tan (3 x) + C$

Step 16

The answer is the antiderivative of the function $f (x) = e^{7 x} + 5 \sec^{2} (3 x)$ .

$F (x) =$ $\frac{1}{7} e^{7 x} + \frac{5}{3} \tan (3 x) + C$