Calculus Examples

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

Step 1

Let $u_{2} = \cos (2 x)$ . Then $d u_{2} = - 2 \sin (2 x) d x$ , so $- \frac{1}{2} d u_{2} = \sin (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 1.1

Let $u_{2} = \cos (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\cos (2 x)$ .

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

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- \sin (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- \sin (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 x) \frac{d}{d x} [x]$

Step 1.1.3.3

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

$- 2 \sin (2 x) \cdot 1$

Step 1.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 x)$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{{u_{2}}^{2}} \cdot \frac{1}{- 2} d u_{2}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

$\int \frac{1}{{u_{2}}^{2}} (- \frac{1}{2}) d u_{2}$

Step 2.2

Multiply $\frac{1}{{u_{2}}^{2}}$ by $\frac{1}{2}$ .

$\int - \frac{1}{{u_{2}}^{2} \cdot 2} d u_{2}$

Step 2.3

Move $2$ to the left of ${u_{2}}^{2}$ .

$\int - \frac{1}{2 {u_{2}}^{2}} d u_{2}$

Step 3

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- \int \frac{1}{2 {u_{2}}^{2}} d u_{2}$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{1}{{u_{2}}^{2}} d u_{2})$

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

Move ${u_{2}}^{2}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{2} \int {({u_{2}}^{2})}^{- 1} d u_{2}$

Step 5.2

Multiply the exponents in ${({u_{2}}^{2})}^{- 1}$ .

Tap for more steps...

Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} \int {u_{2}}^{2 \cdot - 1} d u_{2}$

Step 5.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{2} \int {u_{2}}^{- 2} d u_{2}$

Step 6

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

$- \frac{1}{2} (- \frac{1}{u_{2}}) + C$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.2.2

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

$\frac{1}{2} \cdot \frac{1}{u_{2}} + C$

Step 7.2.3

Multiply $\frac{1}{2}$ by $\frac{1}{u_{2}}$ .

$\frac{1}{2 u_{2}} + C$

Step 8

Replace all occurrences of $u_{2}$ with $\cos (2 x)$ .

$\frac{1}{2 \cos (2 x)} + C$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Separate fractions.

$\frac{1}{2} \cdot \frac{1}{\cos (2 x)} + C$

Step 9.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

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

Step 9.3

Combine $\frac{1}{2}$ and $\sec (2 x)$ .

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

Step 9.4

Reorder terms.

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