Calculus Examples

$8 x$

Step 1

Write $8 x$ as a function.

$f (x) = 8 x$

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.

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

Step 4

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

$8 \int x d x$

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $8 (\frac{1}{2} x^{2} + C)$ as $8 (\frac{1}{2}) x^{2} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $8$ and $\frac{1}{2}$ .

$\frac{8}{2} x^{2} + C$

Step 6.2.2

Cancel the common factor of $8$ and $2$ .

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Step 6.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} x^{2} + C$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1)} x^{2} + C$

Step 6.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} x^{2} + C$

Step 6.2.2.2.3

Rewrite the expression.

$\frac{4}{1} x^{2} + C$

Step 6.2.2.2.4

Divide $4$ by $1$ .

$4 x^{2} + C$

Step 7

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

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