Calculus Examples

$f (x) = 6 \sqrt{x}$

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.

$F (x) = \int 6 \sqrt{x} d x$

Step 3

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

$6 \int \sqrt{x} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$6 \int x^{\frac{1}{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $6 (\frac{2}{3} x^{\frac{3}{2}} + C)$ as $6 (\frac{2}{3}) x^{\frac{3}{2}} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $6$ and $\frac{2}{3}$ .

$\frac{6 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 6.2.2

Multiply $6$ by $2$ .

$\frac{12}{3} x^{\frac{3}{2}} + C$

Step 6.2.3

Cancel the common factor of $12$ and $3$ .

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Step 6.2.3.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3} x^{\frac{3}{2}} + C$

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 4}{3 (1)} x^{\frac{3}{2}} + C$

Step 6.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 \cdot 1} x^{\frac{3}{2}} + C$

Step 6.2.3.2.3

Rewrite the expression.

$\frac{4}{1} x^{\frac{3}{2}} + C$

Step 6.2.3.2.4

Divide $4$ by $1$ .

$4 x^{\frac{3}{2}} + C$

Step 7

The answer is the antiderivative of the function $f (x) = 6 \sqrt{x}$ .

$F (x) =$ $4 x^{\frac{3}{2}} + C$