Calculus Examples

$\int \frac{\sin^{2} (x)}{\cos^{3} (x)} d x$

Step 1

Simplify.

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Step 1.1

Multiply by $1$ .

$\int \frac{\sin^{2} (x) \cdot 1}{\cos^{3} (x)} d x$

Step 1.2

Factor $\cos^{2} (x)$ out of $\cos^{3} (x)$ .

$\int \frac{\sin^{2} (x) \cdot 1}{\cos^{2} (x) \cos (x)} d x$

Step 1.3

Separate fractions.

$\int \frac{1}{\cos (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} d x$

Step 1.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$\int \frac{1}{\cos (x)} \tan^{2} (x) d x$

Step 1.5

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify each term.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

$- (\ln (| \sec (x) + \tan (x) |) + C) + \int \sec (x) \sec^{2} (x) d x$

Step 9

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

$- (\ln (| \sec (x) + \tan (x) |) + C) + \sec (x) \tan (x) - \int \tan (x) (\sec (x) \tan (x)) d x$

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Add $1$ and $1$ .

$- (\ln (| \sec (x) + \tan (x) |) + C) + \sec (x) \tan (x) - \int \tan^{2} (x) \sec (x) d x$

Step 13.2

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

$- (\ln (| \sec (x) + \tan (x) |) + C) + \sec (x) \tan (x) - \int \sec (x) \tan^{2} (x) d x$

Step 14

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

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

Step 15

Simplify by multiplying through.

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Step 15.1

Rewrite the exponentiation as a product.

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

Step 15.2

Apply the distributive property.

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

Step 15.3

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

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

Step 16

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

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

Step 17

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

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

Step 18

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

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

Step 19

Add $1$ and $1$ .

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

Step 20

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

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

Step 21

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

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

Step 22

Add $2$ and $1$ .

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

Step 23

Split the single integral into multiple integrals.

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

Step 24

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

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

Step 25

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

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

Step 26

Simplify by multiplying through.

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Step 26.1

Apply the distributive property.

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

Step 26.2

Multiply $- 1$ by $- 1$ .

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

Step 27

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

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

Step 28

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

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

Step 29

Simplify.

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