Calculus Examples

$\int (\sqrt{x^{3}} - \frac{5}{\sqrt{x}}) d x$

Step 1

Remove parentheses.

$\int \sqrt{x^{3}} - \frac{5}{\sqrt{x}} d x$

Step 2

Split the single integral into multiple integrals.

$\int \sqrt{x^{3}} d x + \int - \frac{5}{\sqrt{x}} d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$\int x^{\frac{3}{2}} d x + \int - \frac{5}{\sqrt{x}} d x$

Step 4

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

$\frac{2}{5} x^{\frac{5}{2}} + C + \int - \frac{5}{\sqrt{x}} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{2}{5} x^{\frac{5}{2}} + C - \int \frac{5}{\sqrt{x}} d x$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 7

Simplify the expression.

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Step 7.1

Multiply $5$ by $- 1$ .

$\frac{2}{5} x^{\frac{5}{2}} + C - 5 \int \frac{1}{\sqrt{x}} d x$

Step 7.2

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 5 \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 7.3

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

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

Step 7.4

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

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Step 7.4.1

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 5 \int x^{\frac{1}{2} \cdot - 1} d x$

Step 7.4.2

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 5 \int x^{\frac{- 1}{2}} d x$

Step 7.4.3

Move the negative in front of the fraction.

$\frac{2}{5} x^{\frac{5}{2}} + C - 5 \int x^{- \frac{1}{2}} d x$

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{2}{5} x^{\frac{5}{2}} - 5 \cdot 2 x^{\frac{1}{2}} + C$

Step 9.2

Multiply $- 5$ by $2$ .

$\frac{2}{5} x^{\frac{5}{2}} - 10 x^{\frac{1}{2}} + C$