Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

$\int \frac{3}{\sqrt{x}} d x + \int x^{2} d x$

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$3 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int x^{2} d x$

Step 5.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 5.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

$6 x^{\frac{1}{2}} + \frac{1}{3} x^{3} + C$

Step 9

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

$F (x) =$ $6 x^{\frac{1}{2}} + \frac{1}{3} x^{3} + C$