Calculus Examples

$\frac{10}{\sqrt{x}} + 10 \sqrt{x}$

Step 1

Split the single integral into multiple integrals.

$\int \frac{10}{\sqrt{x}} d x + \int 10 \sqrt{x} d x$

Step 2

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int \frac{1}{\sqrt{x}} d x + \int 10 \sqrt{x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$10 \int \frac{1}{x^{\frac{1}{2}}} d x + \int 10 \sqrt{x} d x$

Step 3.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$10 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int 10 \sqrt{x} d x$

Step 3.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 3.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$10 \int x^{\frac{1}{2} \cdot - 1} d x + \int 10 \sqrt{x} d x$

Step 3.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$10 \int x^{\frac{- 1}{2}} d x + \int 10 \sqrt{x} d x$

Step 3.3.3

Move the negative in front of the fraction.

$10 \int x^{- \frac{1}{2}} d x + \int 10 \sqrt{x} d x$

Step 4

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

$10 (2 x^{\frac{1}{2}} + C) + \int 10 \sqrt{x} d x$

Step 5

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 (2 x^{\frac{1}{2}} + C) + 10 \int \sqrt{x} d x$

Step 6

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

$10 (2 x^{\frac{1}{2}} + C) + 10 \int x^{\frac{1}{2}} d x$

Step 7

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

$10 (2 x^{\frac{1}{2}} + C) + 10 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$20 x^{\frac{1}{2}} + 10 (\frac{2}{3} x^{\frac{3}{2}}) + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$20 x^{\frac{1}{2}} + 10 \frac{2 x^{\frac{3}{2}}}{3} + C$

Step 8.2.2

Combine $10$ and $\frac{2 x^{\frac{3}{2}}}{3}$ .

$20 x^{\frac{1}{2}} + \frac{10 (2 x^{\frac{3}{2}})}{3} + C$

Step 8.2.3

Multiply $2$ by $10$ .

$20 x^{\frac{1}{2}} + \frac{20 x^{\frac{3}{2}}}{3} + C$

Step 8.3

Reorder terms.

$20 x^{\frac{1}{2}} + \frac{20}{3} x^{\frac{3}{2}} + C$