Calculus Examples

$\int 5 \tan (3 x) \ln (\sec (3 x)) d x$

Step 1

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

$5 \int \tan (3 x) \ln (\sec (3 x)) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\ln (\sec (3 x))$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $\sec (3 x)$ .

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

Step 2.1.3

Rewrite $\sec (3 x)$ in terms of sines and cosines.

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

Step 2.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (3 x)}$ .

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

Step 2.1.5

Multiply $\cos (3 x)$ by $1$ .

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

Step 2.1.6

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 2.1.6.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$\cos (3 x) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [3 x])$

Step 2.1.6.2

The derivative of $\sec (u_{2})$ with respect to $u_{2}$ is $\sec (u_{2}) \tan (u_{2})$ .

$\cos (3 x) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [3 x])$

Step 2.1.6.3

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

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

Step 2.1.7

Differentiate.

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

Simplify the expression.

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Step 2.1.7.3.1

Multiply $3$ by $1$ .

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

Step 2.1.7.3.2

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

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

Step 2.1.8

Simplify.

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Step 2.1.8.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.8.1.1

Add parentheses.

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

Step 2.1.8.1.2

Reorder $\cos (3 x)$ and $\sec (3 x)$ .

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

Step 2.1.8.1.3

Rewrite $3 \cos (3 x) \sec (3 x)$ in terms of sines and cosines.

$3 (\frac{1}{\cos (3 x)} \cos (3 x)) \tan (3 x)$

Step 2.1.8.1.4

Cancel the common factors.

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

Step 2.1.8.2

Multiply $3$ by $1$ .

$3 \tan (3 x)$

Step 2.1.8.3

Rewrite $\tan (3 x)$ in terms of sines and cosines.

$3 \frac{\sin (3 x)}{\cos (3 x)}$

Step 2.1.8.4

Combine $3$ and $\frac{\sin (3 x)}{\cos (3 x)}$ .

$\frac{3 \sin (3 x)}{\cos (3 x)}$

Step 2.1.8.5

Separate fractions.

$\frac{3}{1} \cdot \frac{\sin (3 x)}{\cos (3 x)}$

Step 2.1.8.6

Convert from $\frac{\sin (3 x)}{\cos (3 x)}$ to $\tan (3 x)$ .

$\frac{3}{1} \tan (3 x)$

Step 2.1.8.7

Divide $3$ by $1$ .

$3 \tan (3 x)$

Step 2.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$5 \int u_{3} \frac{1}{3} d u_{3}$

Step 3

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

$5 \int \frac{u_{3}}{3} d u_{3}$

Step 4

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

$5 (\frac{1}{3} \int u_{3} d u_{3})$

Step 5

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

$\frac{5}{3} \int u_{3} d u_{3}$

Step 6

By the Power Rule, the integral of $u_{3}$ with respect to $u_{3}$ is $\frac{1}{2} {u_{3}}^{2}$ .

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

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{5}{3} (\frac{1}{2} {u_{3}}^{2} + C)$ as $\frac{5}{3} \cdot \frac{1}{2} {u_{3}}^{2} + C$ .

$\frac{5}{3} \cdot \frac{1}{2} {u_{3}}^{2} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{5}{3}$ by $\frac{1}{2}$ .

$\frac{5}{3 \cdot 2} {u_{3}}^{2} + C$

Step 7.2.2

Multiply $3$ by $2$ .

$\frac{5}{6} {u_{3}}^{2} + C$

Step 8

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

$\frac{5}{6} \ln^{2} (\sec (3 x)) + C$