Calculus Examples

$x \sqrt[3]{x} + 3 x^{3} \sqrt{x}$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 4.2

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

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Step 4.2.1

Multiply $x$ by $x^{\frac{1}{3}}$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Add $3$ and $1$ .

$\int x^{\frac{4}{3}} + 3 x^{3} \sqrt{x} d x$

Step 4.3

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

$\int x^{\frac{4}{3}} + 3 x^{3} x^{\frac{1}{2}} d x$

Step 4.4

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

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Step 4.4.1

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

$\int x^{\frac{4}{3}} + 3 (x^{\frac{1}{2}} x^{3}) d x$

Step 4.4.2

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

$\int x^{\frac{4}{3}} + 3 x^{\frac{1}{2} + 3} d x$

Step 4.4.3

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

$\int x^{\frac{4}{3}} + 3 x^{\frac{1}{2} + 3 \cdot \frac{2}{2}} d x$

Step 4.4.4

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

$\int x^{\frac{4}{3}} + 3 x^{\frac{1}{2} + \frac{3 \cdot 2}{2}} d x$

Step 4.4.5

Combine the numerators over the common denominator.

$\int x^{\frac{4}{3}} + 3 x^{\frac{1 + 3 \cdot 2}{2}} d x$

Step 4.4.6

Simplify the numerator.

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Step 4.4.6.1

Multiply $3$ by $2$ .

$\int x^{\frac{4}{3}} + 3 x^{\frac{1 + 6}{2}} d x$

Step 4.4.6.2

Add $1$ and $6$ .

$\int x^{\frac{4}{3}} + 3 x^{\frac{7}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int x^{\frac{4}{3}} d x + \int 3 x^{\frac{7}{2}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$\frac{3}{7} x^{\frac{7}{3}} + C + \int 3 x^{\frac{7}{2}} d x$

Step 7

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

$\frac{3}{7} x^{\frac{7}{3}} + C + 3 \int x^{\frac{7}{2}} d x$

Step 8

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

$\frac{3}{7} x^{\frac{7}{3}} + C + 3 (\frac{2}{9} x^{\frac{9}{2}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{3}{7} x^{\frac{7}{3}} + 3 (\frac{2}{9}) x^{\frac{9}{2}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $3$ and $\frac{2}{9}$ .

$\frac{3}{7} x^{\frac{7}{3}} + \frac{3 \cdot 2}{9} x^{\frac{9}{2}} + C$

Step 9.2.2

Multiply $3$ by $2$ .

$\frac{3}{7} x^{\frac{7}{3}} + \frac{6}{9} x^{\frac{9}{2}} + C$

Step 9.2.3

Cancel the common factor of $6$ and $9$ .

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Step 9.2.3.1

Factor $3$ out of $6$ .

$\frac{3}{7} x^{\frac{7}{3}} + \frac{3 (2)}{9} x^{\frac{9}{2}} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $3$ out of $9$ .

$\frac{3}{7} x^{\frac{7}{3}} + \frac{3 \cdot 2}{3 \cdot 3} x^{\frac{9}{2}} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{3}{7} x^{\frac{7}{3}} + \frac{3 \cdot 2}{3 \cdot 3} x^{\frac{9}{2}} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{3}{7} x^{\frac{7}{3}} + \frac{2}{3} x^{\frac{9}{2}} + C$

Step 10

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

$F (x) =$ $\frac{3}{7} x^{\frac{7}{3}} + \frac{2}{3} x^{\frac{9}{2}} + C$