Calculus Examples

$x^{3} \sqrt{x}$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Multiply $x^{3}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1

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

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

Step 4.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $3$ by $2$ .

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

Step 4.2.5.2

Add $6$ and $1$ .

$\int x^{\frac{7}{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{7}{2}}$ with respect to $x$ is $\frac{2}{9} x^{\frac{9}{2}}$ .

$\frac{2}{9} x^{\frac{9}{2}} + C$

Step 6

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

$F (x) =$ $\frac{2}{9} x^{\frac{9}{2}} + C$