Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$3 \int \frac{1}{x^{3}} d x + \int 5 \sqrt{x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$3 \int {(x^{3})}^{- 1} d x + \int 5 \sqrt{x} d x$

Step 3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.1

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

$3 \int x^{3 \cdot - 1} d x + \int 5 \sqrt{x} d x$

Step 3.2.2

Multiply $3$ by $- 1$ .

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

Step 4

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

$3 (- \frac{1}{2} x^{- 2} + C) + \int 5 \sqrt{x} d x$

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 (- \frac{1}{2 x^{2}} + C) + \int 5 \sqrt{x} d x$

Step 6

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

$3 (- \frac{1}{2 x^{2}} + C) + 5 \int \sqrt{x} d x$

Step 7

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

$3 (- \frac{1}{2 x^{2}} + C) + 5 \int x^{\frac{1}{2}} d x$

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Move the negative in front of the fraction.

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

Step 9.2.2

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

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

Step 9.2.3

Combine $5$ and $\frac{2 x^{\frac{3}{2}}}{3}$ .

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

Step 9.2.4

Multiply $2$ by $5$ .

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

Step 9.3

Reorder terms.

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