Calculus Examples

$\int z^{7} (10 z^{8} - 4) d z$

Step 1

Let $u = z^{2}$ . Then $d u = 2 z d z$ , so $\frac{1}{2 z} d u = d z$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = z^{2}$ . Find $\frac{d u}{d z}$ .

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Step 1.1.1

Differentiate $z^{2}$ .

$\frac{d}{d z} [z^{2}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 2$ .

$2 z$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int {\sqrt{u}}^{6} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u}}^{6}$ as $u^{3}$ .

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Step 2.1.1

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

$\int {(u^{\frac{1}{2}})}^{6} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.2

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

$\int u^{\frac{1}{2} \cdot 6} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.3

Combine $\frac{1}{2}$ and $6$ .

$\int u^{\frac{6}{2}} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.4

Cancel the common factor of $6$ and $2$ .

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Step 2.1.4.1

Factor $2$ out of $6$ .

$\int u^{\frac{2 \cdot 3}{2}} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $2$ out of $2$ .

$\int u^{\frac{2 \cdot 3}{2 (1)}} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.4.2.2

Cancel the common factor.

$\int u^{\frac{2 \cdot 3}{2 \cdot 1}} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.4.2.3

Rewrite the expression.

$\int u^{\frac{3}{1}} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.1.4.2.4

Divide $3$ by $1$ .

$\int u^{3} (5 {\sqrt{u}}^{8} - 2) d u$

Step 2.2

Rewrite ${\sqrt{u}}^{8}$ as $u^{4}$ .

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Step 2.2.1

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

$\int u^{3} (5 {(u^{\frac{1}{2}})}^{8} - 2) d u$

Step 2.2.2

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

$\int u^{3} (5 u^{\frac{1}{2} \cdot 8} - 2) d u$

Step 2.2.3

Combine $\frac{1}{2}$ and $8$ .

$\int u^{3} (5 u^{\frac{8}{2}} - 2) d u$

Step 2.2.4

Cancel the common factor of $8$ and $2$ .

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Step 2.2.4.1

Factor $2$ out of $8$ .

$\int u^{3} (5 u^{\frac{2 \cdot 4}{2}} - 2) d u$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $2$ out of $2$ .

$\int u^{3} (5 u^{\frac{2 \cdot 4}{2 (1)}} - 2) d u$

Step 2.2.4.2.2

Cancel the common factor.

$\int u^{3} (5 u^{\frac{2 \cdot 4}{2 \cdot 1}} - 2) d u$

Step 2.2.4.2.3

Rewrite the expression.

$\int u^{3} (5 u^{\frac{4}{1}} - 2) d u$

Step 2.2.4.2.4

Divide $4$ by $1$ .

$\int u^{3} (5 u^{4} - 2) d u$

Step 3

Expand $u^{3} (5 u^{4} - 2)$ .

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Step 3.1

Apply the distributive property.

$\int u^{3} (5 u^{4}) + u^{3} \cdot - 2 d u$

Step 3.2

Reorder $u^{3}$ and $5$ .

$\int 5 \cdot u^{3} u^{4} + u^{3} \cdot - 2 d u$

Step 3.3

Reorder $u^{3}$ and $- 2$ .

$\int 5 \cdot u^{3} u^{4} - 2 \cdot u^{3} d u$

Step 3.4

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

$\int 5 u^{3 + 4} - 2 u^{3} d u$

Step 3.5

Add $3$ and $4$ .

$\int 5 u^{7} - 2 u^{3} d u$

Step 4

Split the single integral into multiple integrals.

$\int 5 u^{7} d u + \int - 2 u^{3} d u$

Step 5

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

$5 \int u^{7} d u + \int - 2 u^{3} d u$

Step 6

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$5 (\frac{1}{8} u^{8} + C) + \int - 2 u^{3} d u$

Step 7

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$5 (\frac{1}{8} u^{8} + C) - 2 \int u^{3} d u$

Step 8

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$5 (\frac{1}{8} u^{8} + C) - 2 (\frac{1}{4} u^{4} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$5 (\frac{1}{8}) u^{8} - 2 (\frac{1}{4}) u^{4} + C$

Step 9.2

Rewrite $5 (\frac{1}{8}) u^{8} - 2 (\frac{1}{4}) u^{4} + C$ as $\frac{5}{8} u^{8} - 2 (\frac{1}{4}) u^{4} + C$ .

$\frac{5}{8} u^{8} - 2 (\frac{1}{4}) u^{4} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $- 2$ and $\frac{1}{4}$ .

$\frac{5}{8} u^{8} + \frac{- 2}{4} u^{4} + C$

Step 9.3.2

Cancel the common factor of $- 2$ and $4$ .

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Step 9.3.2.1

Factor $2$ out of $- 2$ .

$\frac{5}{8} u^{8} + \frac{2 (- 1)}{4} u^{4} + C$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $4$ .

$\frac{5}{8} u^{8} + \frac{2 \cdot - 1}{2 \cdot 2} u^{4} + C$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{5}{8} u^{8} + \frac{2 \cdot - 1}{2 \cdot 2} u^{4} + C$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{5}{8} u^{8} + \frac{- 1}{2} u^{4} + C$

Step 9.3.3

Move the negative in front of the fraction.

$\frac{5}{8} u^{8} - \frac{1}{2} u^{4} + C$

Step 10

Replace all occurrences of $u$ with $z^{2}$ .

$\frac{5}{8} {(z^{2})}^{8} - \frac{1}{2} {(z^{2})}^{4} + C$