Calculus Examples

$\int \sec^{5} (x) d x$

Step 1

Apply the reduction formula.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} \int \sec^{3} (x) d x$

Step 2

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} \int \sec (x) \sec^{2} (x) d x$

Step 3

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \tan (x) (\sec (x) \tan (x)) d x)$

Step 4

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \tan^{1} (x) \tan (x) \sec (x) d x)$

Step 5

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \tan^{1} (x) \tan^{1} (x) \sec (x) d x)$

Step 6

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int {\tan (x)}^{1 + 1} \sec (x) d x)$

Step 7

Simplify the expression.

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Step 7.1

Add $1$ and $1$ .

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \tan^{2} (x) \sec (x) d x)$

Step 7.2

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \sec (x) \tan^{2} (x) d x)$

Step 8

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec^{2} (x)) d x)$

Step 9

Simplify by multiplying through.

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Step 9.1

Rewrite the exponentiation as a product.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \sec (x) (- 1 + \sec (x) \sec (x)) d x)$

Step 9.2

Apply the distributive property.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int \sec (x) \cdot - 1 + \sec (x) (\sec (x) \sec (x)) d x)$

Step 9.3

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \cdot \sec (x) + \sec (x) (\sec (x) \sec (x)) d x)$

Step 10

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec (x) \sec (x) d x)$

Step 11

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{1} (x) \sec^{1} (x) \sec (x) d x)$

Step 12

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{1 + 1} \sec (x) d x)$

Step 13

Add $1$ and $1$ .

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec (x) d x)$

Step 14

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{2} (x) \sec^{1} (x) d x)$

Step 15

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + {\sec (x)}^{2 + 1} d x)$

Step 16

Add $2$ and $1$ .

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - \int - 1 \sec (x) + \sec^{3} (x) d x)$

Step 17

Split the single integral into multiple integrals.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - (\int - 1 \sec (x) d x + \int \sec^{3} (x) d x))$

Step 18

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - (- \int \sec (x) d x + \int \sec^{3} (x) d x))$

Step 19

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - (- (\ln (| \sec (x) + \tan (x) |) + C) + \int \sec^{3} (x) d x))$

Step 20

Simplify by multiplying through.

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Step 20.1

Apply the distributive property.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) - - (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x)$

Step 20.2

Multiply $- 1$ by $- 1$ .

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C) - \int \sec^{3} (x) d x)$

Step 21

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\frac{\sec (x) \tan (x) + 1 (\ln (| \sec (x) + \tan (x) |) + C)}{2} + C)$

Step 22

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

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3}{4} (\frac{\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |) + C}{2} + C)$

Step 23

Simplify.

$\frac{\tan (x) \sec^{3} (x)}{4} + \frac{3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 24

Simplify.

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Step 24.1

To write $\frac{\tan (x) \sec^{3} (x)}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\tan (x) \sec^{3} (x)}{4} \cdot \frac{2}{2} + \frac{3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 24.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 24.2.1

Multiply $\frac{\tan (x) \sec^{3} (x)}{4}$ by $\frac{2}{2}$ .

$\frac{\tan (x) \sec^{3} (x) \cdot 2}{4 \cdot 2} + \frac{3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 24.2.2

Multiply $4$ by $2$ .

$\frac{\tan (x) \sec^{3} (x) \cdot 2}{8} + \frac{3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 24.3

Combine the numerators over the common denominator.

$\frac{\tan (x) \sec^{3} (x) \cdot 2 + 3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 24.4

Move $2$ to the left of $\tan (x) \sec^{3} (x)$ .

$\frac{2 \tan (x) \sec^{3} (x) + 3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))}{8} + C$

Step 25

Reorder terms.

$\frac{1}{8} (2 \tan (x) \sec^{3} (x) + 3 (\sec (x) \tan (x) + \ln (| \sec (x) + \tan (x) |))) + C$