Calculus Examples

$f (x) = 8 x + \sin (x)$

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.

$F (x) = \int 8 x + \sin (x) d x$

Step 3

Split the single integral into multiple integrals.

$\int 8 x d x + \int \sin (x) d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x d x + \int \sin (x) d x$

Step 5

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

$8 (\frac{1}{2} x^{2} + C) + \int \sin (x) d x$

Step 6

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

$8 (\frac{1}{2} x^{2} + C) - \cos (x) + C$

Step 7

Simplify.

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Step 7.1

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

$8 (\frac{x^{2}}{2} + C) - \cos (x) + C$

Step 7.2

Simplify.

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

Step 8

The answer is the antiderivative of the function $f (x) = 8 x + \sin (x)$ .

$F (x) =$ $4 x^{2} - \cos (x) + C$