Calculus Examples

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

Step 1

Write $3 x^{\frac{1}{2}}$ as a function.

$f (x) = 3 x^{\frac{1}{2}}$

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.

$F (x) = \int 3 x^{\frac{1}{2}} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{\frac{1}{2}} d x$

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Multiply $3$ by $2$ .

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

Step 6.2.3

Cancel the common factor of $6$ and $3$ .

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Step 6.2.3.1

Factor $3$ out of $6$ .

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

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $3$ out of $3$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 6.2.3.2.4

Divide $2$ by $1$ .

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

Step 7

The answer is the antiderivative of the function $f (x) = 3 x^{\frac{1}{2}}$ .

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