Calculus Examples

$12 x^{3} \sqrt{x}$

Step 1

Simplify.

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Step 1.1

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

$\int 12 x^{3} x^{\frac{1}{2}} d x$

Step 1.2

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

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Step 1.2.1

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

$\int 12 (x^{\frac{1}{2}} x^{3}) d x$

Step 1.2.2

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

$\int 12 x^{\frac{1}{2} + 3} d x$

Step 1.2.3

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

$\int 12 x^{\frac{1}{2} + 3 \cdot \frac{2}{2}} d x$

Step 1.2.4

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

$\int 12 x^{\frac{1}{2} + \frac{3 \cdot 2}{2}} d x$

Step 1.2.5

Combine the numerators over the common denominator.

$\int 12 x^{\frac{1 + 3 \cdot 2}{2}} d x$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $3$ by $2$ .

$\int 12 x^{\frac{1 + 6}{2}} d x$

Step 1.2.6.2

Add $1$ and $6$ .

$\int 12 x^{\frac{7}{2}} d x$

Step 2

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{\frac{7}{2}} d x$

Step 3

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

$12 (\frac{2}{9} x^{\frac{9}{2}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $12 (\frac{2}{9} x^{\frac{9}{2}} + C)$ as $12 (\frac{2}{9}) x^{\frac{9}{2}} + C$ .

$12 (\frac{2}{9}) x^{\frac{9}{2}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $12$ and $\frac{2}{9}$ .

$\frac{12 \cdot 2}{9} x^{\frac{9}{2}} + C$

Step 4.2.2

Multiply $12$ by $2$ .

$\frac{24}{9} x^{\frac{9}{2}} + C$

Step 4.2.3

Cancel the common factor of $24$ and $9$ .

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Step 4.2.3.1

Factor $3$ out of $24$ .

$\frac{3 (8)}{9} x^{\frac{9}{2}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 8}{3 \cdot 3} x^{\frac{9}{2}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 8}{3 \cdot 3} x^{\frac{9}{2}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{8}{3} x^{\frac{9}{2}} + C$