Calculus Examples

$\int (\frac{z^{6} + 3 z^{4} - z^{2} - 3}{\sqrt{z}}) d z$

Step 1

Remove parentheses.

$\int \frac{z^{6} + 3 z^{4} - z^{2} - 3}{\sqrt{z}} d z$

Step 2

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

$\int \frac{z^{6} + 3 z^{4} - z^{2} - 3}{z^{\frac{1}{2}}} d z$

Step 3

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

$\int (z^{6} + 3 z^{4} - z^{2} - 3) {(z^{\frac{1}{2}})}^{- 1} d z$

Step 4

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

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Step 4.1

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

$\int (z^{6} + 3 z^{4} - z^{2} - 3) z^{\frac{1}{2} \cdot - 1} d z$

Step 4.2

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

$\int (z^{6} + 3 z^{4} - z^{2} - 3) z^{\frac{- 1}{2}} d z$

Step 4.3

Move the negative in front of the fraction.

$\int (z^{6} + 3 z^{4} - z^{2} - 3) z^{- \frac{1}{2}} d z$

Step 5

Expand $(z^{6} + 3 z^{4} - z^{2} - 3) z^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\int (z^{6} + 3 z^{4} - z^{2}) z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.2

Apply the distributive property.

$\int (z^{6} + 3 z^{4}) z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.3

Apply the distributive property.

$\int z^{6} z^{- \frac{1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int z^{6 - \frac{1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int z^{6 \cdot \frac{2}{2} - \frac{1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.6

Combine $6$ and $\frac{2}{2}$ .

$\int z^{\frac{6 \cdot 2}{2} - \frac{1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.7

Combine the numerators over the common denominator.

$\int z^{\frac{6 \cdot 2 - 1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.8

Simplify the numerator.

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Step 5.8.1

Multiply $6$ by $2$ .

$\int z^{\frac{12 - 1}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.8.2

Subtract $1$ from $12$ .

$\int z^{\frac{11}{2}} + 3 z^{4} z^{- \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int z^{\frac{11}{2}} + 3 z^{4 - \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.10

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int z^{\frac{11}{2}} + 3 z^{4 \cdot \frac{2}{2} - \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.11

Combine $4$ and $\frac{2}{2}$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{4 \cdot 2}{2} - \frac{1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.12

Combine the numerators over the common denominator.

$\int z^{\frac{11}{2}} + 3 z^{\frac{4 \cdot 2 - 1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.13

Simplify the numerator.

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Step 5.13.1

Multiply $4$ by $2$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{8 - 1}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.13.2

Subtract $1$ from $8$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{2} z^{- \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.14

Factor out negative.

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - (z^{2} z^{- \frac{1}{2}}) - 3 z^{- \frac{1}{2}} d z$

Step 5.15

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{2 - \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.16

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{2 \cdot \frac{2}{2} - \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.17

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

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{\frac{2 \cdot 2}{2} - \frac{1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.18

Combine the numerators over the common denominator.

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{\frac{2 \cdot 2 - 1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.19

Simplify the numerator.

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Step 5.19.1

Multiply $2$ by $2$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{\frac{4 - 1}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.19.2

Subtract $1$ from $4$ .

$\int z^{\frac{11}{2}} + 3 z^{\frac{7}{2}} - z^{\frac{3}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 5.20

Reorder $z^{\frac{11}{2}}$ and $3 z^{\frac{7}{2}}$ .

$\int 3 z^{\frac{7}{2}} + z^{\frac{11}{2}} - z^{\frac{3}{2}} - 3 z^{- \frac{1}{2}} d z$

Step 6

Split the single integral into multiple integrals.

$\int 3 z^{\frac{7}{2}} d z + \int z^{\frac{11}{2}} d z + \int - z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 7

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

$3 \int z^{\frac{7}{2}} d z + \int z^{\frac{11}{2}} d z + \int - z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 8

By the Power Rule, the integral of $z^{\frac{7}{2}}$ with respect to $z$ is $\frac{2}{9} z^{\frac{9}{2}}$ .

$3 (\frac{2}{9} z^{\frac{9}{2}} + C) + \int z^{\frac{11}{2}} d z + \int - z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 9

By the Power Rule, the integral of $z^{\frac{11}{2}}$ with respect to $z$ is $\frac{2}{13} z^{\frac{13}{2}}$ .

$3 (\frac{2}{9} z^{\frac{9}{2}} + C) + \frac{2}{13} z^{\frac{13}{2}} + C + \int - z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 10

Combine $\frac{2}{9}$ and $z^{\frac{9}{2}}$ .

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C + \int - z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 11

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

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C - \int z^{\frac{3}{2}} d z + \int - 3 z^{- \frac{1}{2}} d z$

Step 12

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

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C - (\frac{2}{5} z^{\frac{5}{2}} + C) + \int - 3 z^{- \frac{1}{2}} d z$

Step 13

Combine $\frac{2}{5}$ and $z^{\frac{5}{2}}$ .

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C - (\frac{2 z^{\frac{5}{2}}}{5} + C) + \int - 3 z^{- \frac{1}{2}} d z$

Step 14

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

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C - (\frac{2 z^{\frac{5}{2}}}{5} + C) - 3 \int z^{- \frac{1}{2}} d z$

Step 15

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

$3 (\frac{2 z^{\frac{9}{2}}}{9} + C) + \frac{2}{13} z^{\frac{13}{2}} + C - (\frac{2 z^{\frac{5}{2}}}{5} + C) - 3 (2 z^{\frac{1}{2}} + C)$

Step 16

Simplify.

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Step 16.1

Simplify.

$\frac{2 z^{\frac{9}{2}}}{3} + \frac{2 z^{\frac{13}{2}}}{13} - \frac{2 z^{\frac{5}{2}}}{5} - 3 (2 z^{\frac{1}{2}}) + C$

Step 16.2

Multiply $2$ by $- 3$ .

$\frac{2 z^{\frac{9}{2}}}{3} + \frac{2 z^{\frac{13}{2}}}{13} - \frac{2 z^{\frac{5}{2}}}{5} - 6 z^{\frac{1}{2}} + C$

Step 17

Reorder terms.

$\frac{2}{3} z^{\frac{9}{2}} + \frac{2}{13} z^{\frac{13}{2}} - \frac{2}{5} z^{\frac{5}{2}} - 6 z^{\frac{1}{2}} + C$