Calculus Examples

$\int 15 x \sqrt{x} + 2 \sqrt{x} d x$

Step 1

Simplify.

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Step 1.1

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

$\int 15 x \cdot x^{\frac{1}{2}} + 2 \sqrt{x} d x$

Step 1.2

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

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Step 1.2.1

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

$\int 15 (x^{\frac{1}{2}} x) + 2 \sqrt{x} d x$

Step 1.2.2

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

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Step 1.2.2.1

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

$\int 15 (x^{\frac{1}{2}} x^{1}) + 2 \sqrt{x} d x$

Step 1.2.2.2

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

$\int 15 x^{\frac{1}{2} + 1} + 2 \sqrt{x} d x$

Step 1.2.3

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

$\int 15 x^{\frac{1}{2} + \frac{2}{2}} + 2 \sqrt{x} d x$

Step 1.2.4

Combine the numerators over the common denominator.

$\int 15 x^{\frac{1 + 2}{2}} + 2 \sqrt{x} d x$

Step 1.2.5

Add $1$ and $2$ .

$\int 15 x^{\frac{3}{2}} + 2 \sqrt{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 15 x^{\frac{3}{2}} d x + \int 2 \sqrt{x} d x$

Step 3

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$15 \int x^{\frac{3}{2}} d x + \int 2 \sqrt{x} d x$

Step 4

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

$15 (\frac{2}{5} x^{\frac{5}{2}} + C) + \int 2 \sqrt{x} d x$

Step 5

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

$15 (\frac{2}{5} x^{\frac{5}{2}} + C) + 2 \int \sqrt{x} d x$

Step 6

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

$15 (\frac{2}{5} x^{\frac{5}{2}} + C) + 2 \int x^{\frac{1}{2}} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$15 (\frac{2}{5} x^{\frac{5}{2}} + C) + 2 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$15 (\frac{2}{5}) x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $15$ and $\frac{2}{5}$ .

$\frac{15 \cdot 2}{5} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.2

Multiply $15$ by $2$ .

$\frac{30}{5} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.3

Cancel the common factor of $30$ and $5$ .

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Step 8.2.3.1

Factor $5$ out of $30$ .

$\frac{5 \cdot 6}{5} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 6}{5 (1)} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 6}{5 \cdot 1} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.3.2.3

Rewrite the expression.

$\frac{6}{1} x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.3.2.4

Divide $6$ by $1$ .

$6 x^{\frac{5}{2}} + 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 8.2.4

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

$6 x^{\frac{5}{2}} + \frac{2 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 8.2.5

Multiply $2$ by $2$ .

$6 x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C$