Calculus Examples

\sec^{3} (x)

$\sec^{3} (x)$

Step 1

Factor

\sec (x)

$\sec (x)$ out of

\sec^{3} (x)

$\sec^{3} (x)$ .

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

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

Step 2

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = \sec (x)

$u = \sec (x)$ and

d v = \sec^{2} (x)

$d v = \sec^{2} (x)$ .

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

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

Step 3

Raise

\tan (x)

$\tan (x)$ to the power of

1

$1$ .

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

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

Step 4

Raise

\tan (x)

$\tan (x)$ to the power of

1

$1$ .

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

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

Step 5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 6

Simplify the expression.

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Step 6.1

Add

1

$1$ and

1

$1$ .

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

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

Step 6.2

Reorder

\tan^{2} (x)

$\tan^{2} (x)$ and

\sec (x)

$\sec (x)$ .

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

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

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

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

Step 7

Using the Pythagorean Identity, rewrite

\tan^{2} (x)

$\tan^{2} (x)$ as

- 1 + \sec^{2} (x)

$- 1 + \sec^{2} (x)$ .

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

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

Step 8

Simplify by multiplying through.

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Step 8.1

Rewrite the exponentiation as a product.

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

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

Step 8.2

Apply the distributive property.

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

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

Step 8.3

Reorder

\sec (x)

$\sec (x)$ and

- 1

$- 1$ .

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

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

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

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

Step 9

Raise

\sec (x)

$\sec (x)$ to the power of

1

$1$ .

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

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

Step 10

Raise

\sec (x)

$\sec (x)$ to the power of

1

$1$ .

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

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

Step 11

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 12

Add

1

$1$ and

1

$1$ .

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

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

Step 13

Raise

\sec (x)

$\sec (x)$ to the power of

1

$1$ .

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

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

Step 14

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 15

Add

2

$2$ and

1

$1$ .

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

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

Step 16

Split the single integral into multiple integrals.

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

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

Step 17

Since

- 1

$- 1$ is constant with respect to

x

$x$ , move

- 1

$- 1$ out of the integral.

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

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

Step 18

The integral of

\sec (x)

$\sec (x)$ with respect to

x

$x$ is

\ln (| \sec (x) + \tan (x) |)

$\ln (| \sec (x) + \tan (x) |)$ .

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

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

Step 19

Simplify by multiplying through.

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Step 19.1

Apply the distributive property.

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

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

Step 19.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

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

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

Step 20

Solving for

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

$\int \sec^{3} (x) d x$ , we find that

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

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

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

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

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

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

Step 21

Multiply

\ln (| \sec (x) + \tan (x) |) + C

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

1

$1$ .

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

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

Step 22

Simplify.

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

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