Calculus Examples

$\int \sqrt{r} - \frac{1}{r^{3}} d r$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt{r} d r + \int - \frac{1}{r^{3}} d r$

Step 2

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

$\int r^{\frac{1}{2}} d r + \int - \frac{1}{r^{3}} d r$

Step 3

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

$\frac{2}{3} r^{\frac{3}{2}} + C + \int - \frac{1}{r^{3}} d r$

Step 4

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

$\frac{2}{3} r^{\frac{3}{2}} + C - \int \frac{1}{r^{3}} d r$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

$\frac{2}{3} r^{\frac{3}{2}} + C - \int r^{3 \cdot - 1} d r$

Step 5.2.2

Multiply $3$ by $- 1$ .

$\frac{2}{3} r^{\frac{3}{2}} + C - \int r^{- 3} d r$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Simplify.

$\frac{2}{3} r^{\frac{3}{2}} - - \frac{1}{2 r^{2}} + C$

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2}{3} r^{\frac{3}{2}} + 1 \frac{1}{2 r^{2}} + C$

Step 7.3.2

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

$\frac{2}{3} r^{\frac{3}{2}} + \frac{1}{2 r^{2}} + C$