Calculus Examples

$f (x) = 2 \sin (6 x + 3) - 4 x$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 2 \sin (6 x + 3) - 4 x d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 \sin (6 x + 3) d x + \int - 4 x d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sin (6 x + 3) d x + \int - 4 x d x$

Step 5

Let $u = 6 x + 3$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = 6 x + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 5.1.1

Differentiate $6 x + 3$ .

$\frac{d}{d x} [6 x + 3]$

Step 5.1.2

By the Sum Rule, the derivative of $6 x + 3$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [3]$

Step 5.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

Tap for more steps...

Step 5.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 5.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [3]$

Step 5.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [3]$

Step 5.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 5.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$6 + 0$

Step 5.1.4.2

Add $6$ and $0$ .

$6$

Step 5.2

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

$2 \int \sin (u) \frac{1}{6} d u + \int - 4 x d x$

Step 6

Combine $\sin (u)$ and $\frac{1}{6}$ .

$2 \int \frac{\sin (u)}{6} d u + \int - 4 x d x$

Step 7

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$2 (\frac{1}{6} \int \sin (u) d u) + \int - 4 x d x$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Combine $\frac{1}{6}$ and $2$ .

$\frac{2}{6} \int \sin (u) d u + \int - 4 x d x$

Step 8.2

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 8.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} \int \sin (u) d u + \int - 4 x d x$

Step 8.2.2

Cancel the common factors.

Tap for more steps...

Step 8.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} \int \sin (u) d u + \int - 4 x d x$

Step 8.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} \int \sin (u) d u + \int - 4 x d x$

Step 8.2.2.3

Rewrite the expression.

$\frac{1}{3} \int \sin (u) d u + \int - 4 x d x$

Step 9

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

$\frac{1}{3} (- \cos (u) + C) + \int - 4 x d x$

Step 10

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

$\frac{1}{3} (- \cos (u) + C) - 4 \int x d x$

Step 11

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

$\frac{1}{3} (- \cos (u) + C) - 4 (\frac{1}{2} x^{2} + C)$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

$- \frac{\cos (u)}{3} - 4 (\frac{1}{2} x^{2}) + C$

Step 12.2

Simplify.

Tap for more steps...

Step 12.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$- \frac{\cos (u)}{3} - 4 \frac{x^{2}}{2} + C$

Step 12.2.2

Combine $- 4$ and $\frac{x^{2}}{2}$ .

$- \frac{\cos (u)}{3} + \frac{- 4 x^{2}}{2} + C$

Step 12.2.3

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 12.2.3.1

Factor $2$ out of $- 4 x^{2}$ .

$- \frac{\cos (u)}{3} + \frac{2 (- 2 x^{2})}{2} + C$

Step 12.2.3.2

Cancel the common factors.

Tap for more steps...

Step 12.2.3.2.1

Factor $2$ out of $2$ .

$- \frac{\cos (u)}{3} + \frac{2 (- 2 x^{2})}{2 (1)} + C$

Step 12.2.3.2.2

Cancel the common factor.

$- \frac{\cos (u)}{3} + \frac{2 (- 2 x^{2})}{2 \cdot 1} + C$

Step 12.2.3.2.3

Rewrite the expression.

$- \frac{\cos (u)}{3} + \frac{- 2 x^{2}}{1} + C$

Step 12.2.3.2.4

Divide $- 2 x^{2}$ by $1$ .

$- \frac{\cos (u)}{3} - 2 x^{2} + C$

Step 13

Replace all occurrences of $u$ with $6 x + 3$ .

$- \frac{\cos (6 x + 3)}{3} - 2 x^{2} + C$

Step 14

Reorder terms.

$- \frac{1}{3} \cos (6 x + 3) - 2 x^{2} + C$

Step 15

The answer is the antiderivative of the function $f (x) = 2 \sin (6 x + 3) - 4 x$ .

$F (x) =$ $- \frac{1}{3} \cos (6 x + 3) - 2 x^{2} + C$