Calculus Examples

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

Step 1

Write $\tan^{2} (2 x)$ as a function.

$f (x) = \tan^{2} (2 x)$

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.

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

Step 4

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 4.1.2

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

$2 \frac{d}{d x} [x]$

Step 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

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

Step 5

Combine $\tan^{2} (u)$ and $\frac{1}{2}$ .

$\int \frac{\tan^{2} (u)}{2} d u$

Step 6

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

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

Step 7

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

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Apply the constant rule.

$\frac{1}{2} (- u + C + \int \sec^{2} (u) d u)$

Step 10

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

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

Step 11

Simplify.

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

Step 12

Replace all occurrences of $u$ with $2 x$ .

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

Multiply $2$ by $- 1$ .

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

Step 13.2

Apply the distributive property.

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

Step 13.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.3.1

Factor $2$ out of $- 2 x$ .

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

Step 13.3.2

Cancel the common factor.

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

Step 13.3.3

Rewrite the expression.

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

Step 13.4

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

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

Step 14

Reorder terms.

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

Step 15

The answer is the antiderivative of the function $f (x) = \tan^{2} (2 x)$ .

$F (x) =$ $- x + \frac{1}{2} \tan (2 x) + C$