Calculus Examples

$4 \sec (3 x) \tan (3 x)$

Step 1

Write $4 \sec (3 x) \tan (3 x)$ as a function.

$f (x) = 4 \sec (3 x) \tan (3 x)$

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.

$F (x) = \int 4 \sec (3 x) \tan (3 x) d x$

Step 4

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

$4 \int \sec (3 x) \tan (3 x) d x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $\sec (3 x)$ .

$\frac{d}{d x} [\sec (3 x)]$

Step 5.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 3 x$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [3 x]$

Step 5.1.2.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [3 x]$

Step 5.1.2.3

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

$\sec (3 x) \tan (3 x) \frac{d}{d x} [3 x]$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

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

$\sec (3 x) \tan (3 x) (3 \frac{d}{d x} [x])$

Step 5.1.3.2

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

$\sec (3 x) \tan (3 x) (3 \cdot 1)$

Step 5.1.3.3

Simplify the expression.

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Step 5.1.3.3.1

Multiply $3$ by $1$ .

$\sec (3 x) \tan (3 x) \cdot 3$

Step 5.1.3.3.2

Move $3$ to the left of $\sec (3 x) \tan (3 x)$ .

$3 \sec (3 x) \tan (3 x)$

Step 5.2

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

$4 \int \frac{1}{3} d u_{2}$

Step 6

Apply the constant rule.

$4 (\frac{1}{3} u_{2} + C)$

Step 7

Simplify the answer.

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Step 7.1

Rewrite $4 (\frac{1}{3} u_{2} + C)$ as $4 (\frac{1}{3}) u_{2} + C$ .

$4 (\frac{1}{3}) u_{2} + C$

Step 7.2

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

$\frac{4}{3} u_{2} + C$

Step 7.3

Replace all occurrences of $u_{2}$ with $\sec (3 x)$ .

$\frac{4}{3} \sec (3 x) + C$

Step 8

The answer is the antiderivative of the function $f (x) = 4 \sec (3 x) \tan (3 x)$ .

$F (x) =$ $\frac{4}{3} \sec (3 x) + C$