Calculus Examples

$f (x) = 6 x$

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.

$F (x) = \int 6 x d x$

Step 3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $6 (\frac{1}{2} x^{2} + C)$ as $6 (\frac{1}{2}) x^{2} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{6}{2} x^{2} + C$

Step 5.2.2

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

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Step 5.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2} x^{2} + C$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 1} x^{2} + C$

Step 5.2.2.2.3

Rewrite the expression.

$\frac{3}{1} x^{2} + C$

Step 5.2.2.2.4

Divide $3$ by $1$ .

$3 x^{2} + C$

Step 6

The answer is the antiderivative of the function $f (x) = 6 x$ .

$F (x) =$ $3 x^{2} + C$