Calculus Examples

$x - \sin (2 x)$

Step 1

Write $x - \sin (2 x)$ as a function.

$f (x) = x - \sin (2 x)$

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.

$F (x) = \int x - \sin (2 x) d x$

Step 4

Split the single integral into multiple integrals.

$\int x d x + \int - \sin (2 x) d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2} + C + \int - \sin (2 x) d x$

Step 6

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

$\frac{1}{2} x^{2} + C - \int \sin (2 x) d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

Multiply $- 1$ by $- 1$ .

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

Step 11.2.2

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

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

Step 11.2.3

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

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

Step 12

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

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

Step 13

Reorder terms.

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

Step 14

The answer is the antiderivative of the function $f (x) = x - \sin (2 x)$ .

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