Calculus Examples

$\int (\frac{3 w^{2}}{2} - \frac{2}{3 w^{2}}) d w$

Step 1

Remove parentheses.

$\int \frac{3 w^{2}}{2} - \frac{2}{3 w^{2}} d w$

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $\frac{3}{2}$ is constant with respect to $w$ , move $\frac{3}{2}$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

Since $\frac{2}{3}$ is constant with respect to $w$ , move $\frac{2}{3}$ out of the integral.

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

Step 7

Apply basic rules of exponents.

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Step 7.1

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

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

Step 7.2

Multiply the exponents in ${(w^{2})}^{- 1}$ .

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Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{3}$ .

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

Step 9.2.2

Multiply $2$ by $3$ .

$\frac{3}{6} w^{3} - \frac{2}{3} (- w^{- 1}) + C$

Step 9.2.3

Cancel the common factor of $3$ and $6$ .

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Step 9.2.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6} w^{3} - \frac{2}{3} (- w^{- 1}) + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $3$ out of $6$ .

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

Step 9.2.3.2.2

Cancel the common factor.

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

Step 9.2.3.2.3

Rewrite the expression.

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

Step 9.2.4

Multiply $- 1$ by $- 1$ .

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

Step 9.2.5

Multiply $\frac{2}{3}$ by $1$ .

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

Step 9.2.6

Combine $\frac{2}{3}$ and $w^{- 1}$ .

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

Step 9.2.7

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

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