Calculus Examples

$\int_{0}^{1} {(8 y^{2} - y + 1)}^{- \frac{1}{3}} (32 y - 2) d y$

Step 1

Simplify.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int_{0}^{1} \frac{1}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} (32 y - 2) d y$

Step 1.2

Multiply $\frac{1}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}}$ by $32 y - 2$ .

$\int_{0}^{1} \frac{32 y - 2}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} d y$

Step 1.3

Factor $2$ out of $32 y - 2$ .

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Step 1.3.1

Factor $2$ out of $32 y$ .

$\int_{0}^{1} \frac{2 (16 y) - 2}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} d y$

Step 1.3.2

Factor $2$ out of $- 2$ .

$\int_{0}^{1} \frac{2 (16 y) + 2 (- 1)}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} d y$

Step 1.3.3

Factor $2$ out of $2 (16 y) + 2 (- 1)$ .

$\int_{0}^{1} \frac{2 (16 y - 1)}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} d y$

Step 2

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

$2 \int_{0}^{1} \frac{16 y - 1}{{(8 y^{2} - y + 1)}^{\frac{1}{3}}} d y$

Step 3

Let $u = 8 y^{2} - y + 1$ . Then $d u = (16 y - 1) d y$ , so $\frac{1}{16 y - 1} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 8 y^{2} - y + 1$ . Find $\frac{d u}{d y}$ .

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Step 3.1.1

Differentiate $8 y^{2} - y + 1$ .

$\frac{d}{d y} [8 y^{2} - y + 1]$

Step 3.1.2

By the Sum Rule, the derivative of $8 y^{2} - y + 1$ with respect to $y$ is $\frac{d}{d y} [8 y^{2}] + \frac{d}{d y} [- y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [8 y^{2}] + \frac{d}{d y} [- y] + \frac{d}{d y} [1]$

Step 3.1.3

Evaluate $\frac{d}{d y} [8 y^{2}]$ .

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Step 3.1.3.1

Since $8$ is constant with respect to $y$ , the derivative of $8 y^{2}$ with respect to $y$ is $8 \frac{d}{d y} [y^{2}]$ .

$8 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- y] + \frac{d}{d y} [1]$

Step 3.1.3.2

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

$8 (2 y) + \frac{d}{d y} [- y] + \frac{d}{d y} [1]$

Step 3.1.3.3

Multiply $2$ by $8$ .

$16 y + \frac{d}{d y} [- y] + \frac{d}{d y} [1]$

Step 3.1.4

Evaluate $\frac{d}{d y} [- y]$ .

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Step 3.1.4.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

$16 y - \frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 3.1.4.2

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

$16 y - 1 \cdot 1 + \frac{d}{d y} [1]$

Step 3.1.4.3

Multiply $- 1$ by $1$ .

$16 y - 1 + \frac{d}{d y} [1]$

Step 3.1.5

Differentiate using the Constant Rule.

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Step 3.1.5.1

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$16 y - 1 + 0$

Step 3.1.5.2

Add $16 y - 1$ and $0$ .

$16 y - 1$

Step 3.2

Substitute the lower limit in for $y$ in $u = 8 y^{2} - y + 1$ .

$u_{lower} = 8 \cdot 0^{2} - 0 + 1$

Step 3.3

Simplify.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 8 \cdot 0 - 0 + 1$

Step 3.3.1.2

Multiply $8$ by $0$ .

$u_{lower} = 0 - 0 + 1$

Step 3.3.1.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0 + 0 + 1$

Step 3.3.2

Add $0$ and $0$ .

$u_{lower} = 0 + 1$

Step 3.3.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 3.4

Substitute the upper limit in for $y$ in $u = 8 y^{2} - y + 1$ .

$u_{upper} = 8 \cdot 1^{2} - 1 \cdot 1 + 1$

Step 3.5

Simplify.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

One to any power is one.

$u_{upper} = 8 \cdot 1 - 1 \cdot 1 + 1$

Step 3.5.1.2

Multiply $8$ by $1$ .

$u_{upper} = 8 - 1 \cdot 1 + 1$

Step 3.5.1.3

Multiply $- 1$ by $1$ .

$u_{upper} = 8 - 1 + 1$

Step 3.5.2

Subtract $1$ from $8$ .

$u_{upper} = 7 + 1$

Step 3.5.3

Add $7$ and $1$ .

$u_{upper} = 8$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 8$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$2 \int_{1}^{8} \frac{1}{u^{\frac{1}{3}}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 4.2.1

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

$2 \int_{1}^{8} u^{\frac{1}{3} \cdot - 1} d u$

Step 4.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$2 \int_{1}^{8} u^{\frac{- 1}{3}} d u$

Step 4.2.3

Move the negative in front of the fraction.

$2 \int_{1}^{8} u^{- \frac{1}{3}} d u$

Step 5

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

$2 (\frac{3}{2} u^{\frac{2}{3}}]_{1}^{8})$

Step 6

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

$2 (\frac{3 u^{\frac{2}{3}}}{2}]_{1}^{8})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{3 u^{\frac{2}{3}}}{2}$ at $8$ and at $1$ .

$2 ((\frac{3 \cdot 8^{\frac{2}{3}}}{2}) - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2

Simplify.

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Step 7.2.1

Rewrite $8$ as $2^{3}$ .

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor of $3$ .

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Step 7.2.3.1

Cancel the common factor.

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

Step 7.2.3.2

Rewrite the expression.

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

Step 7.2.4

Raise $2$ to the power of $2$ .

$2 (\frac{3 \cdot 4}{2} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.5

Multiply $3$ by $4$ .

$2 (\frac{12}{2} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.6

Cancel the common factor of $12$ and $2$ .

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Step 7.2.6.1

Factor $2$ out of $12$ .

$2 (\frac{2 \cdot 6}{2} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.6.2

Cancel the common factors.

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Step 7.2.6.2.1

Factor $2$ out of $2$ .

$2 (\frac{2 \cdot 6}{2 (1)} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.6.2.2

Cancel the common factor.

$2 (\frac{2 \cdot 6}{2 \cdot 1} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.6.2.3

Rewrite the expression.

$2 (\frac{6}{1} - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.6.2.4

Divide $6$ by $1$ .

$2 (6 - \frac{3 \cdot 1^{\frac{2}{3}}}{2})$

Step 7.2.7

One to any power is one.

$2 (6 - \frac{3 \cdot 1}{2})$

Step 7.2.8

Multiply $3$ by $1$ .

$2 (6 - \frac{3}{2})$

Step 7.2.9

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

$2 (6 \cdot \frac{2}{2} - \frac{3}{2})$

Step 7.2.10

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

$2 (\frac{6 \cdot 2}{2} - \frac{3}{2})$

Step 7.2.11

Combine the numerators over the common denominator.

$2 \frac{6 \cdot 2 - 3}{2}$

Step 7.2.12

Simplify the numerator.

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Step 7.2.12.1

Multiply $6$ by $2$ .

$2 \frac{12 - 3}{2}$

Step 7.2.12.2

Subtract $3$ from $12$ .

$2 (\frac{9}{2})$

Step 7.2.13

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

$\frac{2 \cdot 9}{2}$

Step 7.2.14

Multiply $2$ by $9$ .

$\frac{18}{2}$

Step 7.2.15

Cancel the common factor of $18$ and $2$ .

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Step 7.2.15.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9}{2}$

Step 7.2.15.2

Cancel the common factors.

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Step 7.2.15.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 9}{2 (1)}$

Step 7.2.15.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 1}$

Step 7.2.15.2.3

Rewrite the expression.

$\frac{9}{1}$

Step 7.2.15.2.4

Divide $9$ by $1$ .

$9$

Step 8