Calculus Examples

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

Step 1

Write $\frac{1}{\cos^{3} (x)}$ as a function.

$f (x) = \frac{1}{\cos^{3} (x)}$

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.

$F (x) = \int \frac{1}{\cos^{3} (x)} d x$

Step 4

Convert from $\frac{1}{\cos^{3} (x)}$ to $\sec^{3} (x)$ .

$\int \sec^{3} (x) d x$

Step 5

Factor $\sec (x)$ out of $\sec^{3} (x)$ .

$\int \sec (x) \sec^{2} (x) d x$

Step 6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sec (x)$ and $d v = \sec^{2} (x)$ .

$\sec (x) \tan (x) - \int \tan (x) (\sec (x) \tan (x)) d x$

Step 7

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) \tan (x) - \int \tan^{1} (x) \tan (x) \sec (x) d x$

Step 8

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) \tan (x) - \int \tan^{1} (x) \tan^{1} (x) \sec (x) d x$

Step 9

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

$\sec (x) \tan (x) - \int {\tan (x)}^{1 + 1} \sec (x) d x$

Step 10

Simplify the expression.

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Step 10.1

Add $1$ and $1$ .

$\sec (x) \tan (x) - \int \tan^{2} (x) \sec (x) d x$

Step 10.2

Reorder $\tan^{2} (x)$ and $\sec (x)$ .

$\sec (x) \tan (x) - \int \sec (x) \tan^{2} (x) d x$

Step 11

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec^{2} (x)) d x$

Step 12

Simplify by multiplying through.

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Step 12.1

Rewrite the exponentiation as a product.

$\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec (x) \sec (x)) d x$

Step 12.2

Apply the distributive property.

$\sec (x) \tan (x) - \int \sec (x) \cdot - 1 + \sec (x) (\sec (x) \sec (x)) d x$

Step 12.3

Reorder $\sec (x)$ and $- 1$ .

$\sec (x) \tan (x) - \int - 1 \cdot \sec (x) + \sec (x) (\sec (x) \sec (x)) d x$

Step 13

Raise $\sec (x)$ to the power of $1$ .

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec (x) \sec (x) d x$

Step 14

Raise $\sec (x)$ to the power of $1$ .

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec^{1} (x) \sec (x) d x$

Step 15

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

$\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{1 + 1} \sec (x) d x$

Step 16

Add $1$ and $1$ .

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec (x) d x$

Step 17

Raise $\sec (x)$ to the power of $1$ .

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec^{1} (x) d x$

Step 18

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

$\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{2 + 1} d x$

Step 19

Add $2$ and $1$ .

$\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{3} (x) d x$

Step 20

Split the single integral into multiple integrals.

$\sec (x) \tan (x) - (\int - 1 \sec (x) d x + \int \sec^{3} (x) d x)$

Step 21

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

$\sec (x) \tan (x) - (- \int \sec (x) d x + \int \sec^{3} (x) d x)$

Step 22

The integral of $\sec (x)$ with respect to $x$ is $\ln (| \sec (x) + \tan (x) |)$ .

$\sec (x) \tan (x) - (- (\ln (| \sec (x) + \tan (x) |) + C) + \int \sec^{3} (x) d x)$

Step 23

Simplify by multiplying through.

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Step 23.1

Apply the distributive property.

$\sec (x) \tan (x) - - (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x$

Step 23.2

Multiply $- 1$ by $- 1$ .

$\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x$

Step 24

Solving for $\int \sec^{3} (x) d x$ , we find that $\int \sec^{3} (x) d x$ = $\frac{\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C)}{2}$ .

$\frac{\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C)}{2} + C$

Step 25

Multiply $\ln (| \sec (x) + \tan (x) |) + C$ by $1$ .

$\frac{\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |) + C}{2} + C$

Step 26

Simplify.

$\frac{1}{2} (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |)) + C$

Step 27

The answer is the antiderivative of the function $f (x) = \frac{1}{\cos^{3} (x)}$ .

$F (x) =$ $\frac{1}{2} (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |)) + C$