Calculus Examples

$\int 2 - \tan (\frac{x}{4}) d x$

Step 1

Split the single integral into multiple integrals.

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

Step 2

Apply the constant rule.

$2 x + C + \int - \tan (\frac{x}{4}) d x$

Step 3

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

$2 x + C - \int \tan (\frac{x}{4}) d x$

Step 4

Let $u = \frac{x}{4}$ . Then $d u = \frac{1}{4} d x$ , so $4 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \frac{x}{4}$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\frac{x}{4}$ .

$\frac{d}{d x} [\frac{x}{4}]$

Step 4.1.2

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$\frac{1}{4} \frac{d}{d x} [x]$

Step 4.1.3

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

$\frac{1}{4} \cdot 1$

Step 4.1.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4}$

Step 4.2

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

$2 x + C - \int \tan (u) \frac{1}{\frac{1}{4}} d u$

Step 5

Simplify.

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Step 5.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{4}$ .

$2 x + C - \int \tan (u) (1 \cdot 4) d u$

Step 5.2

Multiply $4$ by $1$ .

$2 x + C - \int \tan (u) \cdot 4 d u$

Step 5.3

Move $4$ to the left of $\tan (u)$ .

$2 x + C - \int 4 \tan (u) d u$

Step 6

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

$2 x + C - (4 \int \tan (u) d u)$

Step 7

Multiply $4$ by $- 1$ .

$2 x + C - 4 \int \tan (u) d u$

Step 8

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

$2 x + C - 4 (\ln (| \sec (u) |) + C)$

Step 9

Simplify.

$2 x - 4 \ln (| \sec (u) |) + C$

Step 10

Replace all occurrences of $u$ with $\frac{x}{4}$ .

$2 x - 4 \ln (| \sec (\frac{x}{4}) |) + C$

Step 11

Reorder terms.

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