Calculus Examples

$f (x) = x \sqrt{x} - \frac{3}{\sqrt{x}}$

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.

$F (x) = \int x \sqrt{x} - \frac{3}{\sqrt{x}} d x$

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.2.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 3.2.1.1

Raise $x$ to the power of $1$ .

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

Step 3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $2$ and $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + \int - \frac{3}{\sqrt{x}} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{2}{5} x^{\frac{5}{2}} + C - \int \frac{3}{\sqrt{x}} d x$

Step 7

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

$\frac{2}{5} x^{\frac{5}{2}} + C - (3 \int \frac{1}{\sqrt{x}} d x)$

Step 8

Simplify the expression.

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Step 8.1

Multiply $3$ by $- 1$ .

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int \frac{1}{\sqrt{x}} d x$

Step 8.2

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 8.3

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 8.4

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

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Step 8.4.1

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int x^{\frac{1}{2} \cdot - 1} d x$

Step 8.4.2

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int x^{\frac{- 1}{2}} d x$

Step 8.4.3

Move the negative in front of the fraction.

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 \int x^{- \frac{1}{2}} d x$

Step 9

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 3 (2 x^{\frac{1}{2}} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{2}{5} x^{\frac{5}{2}} - 3 \cdot 2 x^{\frac{1}{2}} + C$

Step 10.2

Multiply $- 3$ by $2$ .

$\frac{2}{5} x^{\frac{5}{2}} - 6 x^{\frac{1}{2}} + C$

Step 11

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

$F (x) =$ $\frac{2}{5} x^{\frac{5}{2}} - 6 x^{\frac{1}{2}} + C$