Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Factor $x^{2}$ out of $x^{3} + 2 x^{2}$ .

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Step 5.1.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 5.1.2

Factor $x^{2}$ out of $2 x^{2}$ .

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

Step 5.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 2$ .

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

Step 5.2

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 5.3

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

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Step 5.3.1

Move $x^{- \frac{1}{2}}$ .

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

Step 5.3.2

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

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

Step 5.3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.4

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

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

Step 5.3.5

Combine the numerators over the common denominator.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Multiply $2$ by $2$ .

$\int x^{\frac{- 1 + 4}{2}} (x + 2) d x$

Step 5.3.6.2

Add $- 1$ and $4$ .

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Reorder $x^{\frac{3}{2}}$ and $2$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Combine the numerators over the common denominator.

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

Step 6.7

Add $3$ and $2$ .

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

Step 6.8

Reorder $x^{\frac{5}{2}}$ and $2 x^{\frac{3}{2}}$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 9

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

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

Step 10

By the Power Rule, the integral of $x^{\frac{5}{2}}$ with respect to $x$ is $\frac{2}{7} x^{\frac{7}{2}}$ .

$2 (\frac{2}{5} x^{\frac{5}{2}} + C) + \frac{2}{7} x^{\frac{7}{2}} + C$

Step 11

Simplify.

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Step 11.1

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$2 (\frac{2 x^{\frac{5}{2}}}{5} + C) + \frac{2}{7} x^{\frac{7}{2}} + C$

Step 11.2

Simplify.

$\frac{4 x^{\frac{5}{2}}}{5} + \frac{2}{7} x^{\frac{7}{2}} + C$

Step 11.3

Reorder terms.

$\frac{4}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}} + C$

Step 12

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

$F (x) =$ $\frac{4}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}} + C$