Calculus Examples

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

Step 1

Let $u = 3 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 - y$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $3 - y$ .

$\frac{d}{d y} [3 - y]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 - y$ with respect to $y$ is $\frac{d}{d y} [3] + \frac{d}{d y} [- y]$ .

$\frac{d}{d y} [3] + \frac{d}{d y} [- y]$

Step 1.1.2.2

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- y]$

Step 1.1.3

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

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Step 1.1.3.1

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

$0 - \frac{d}{d y} [y]$

Step 1.1.3.2

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

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $u^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(u^{\frac{2}{3}})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 3

Expand $(- (- u + 3) - 3) u^{- \frac{2}{3}}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Move parentheses.

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

Step 3.5

Multiply $- 1$ by $- 1$ .

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

Step 3.6

Multiply $u$ by $1$ .

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

Step 3.7

Raise $u$ to the power of $1$ .

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

Step 3.8

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

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

Step 3.9

Write $1$ as a fraction with a common denominator.

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Subtract $2$ from $3$ .

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

Step 3.12

Multiply $- 1$ by $3$ .

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

Step 3.13

Subtract $3 u^{- \frac{2}{3}}$ from $- 3 u^{- \frac{2}{3}}$ .

$\int u^{\frac{1}{3}} - 6 u^{- \frac{2}{3}} d u$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\frac{3}{4} u^{\frac{4}{3}} + C + \int - 6 u^{- \frac{2}{3}} d u$

Step 6

Since $- 6$ is constant with respect to $u$ , move $- 6$ out of the integral.

$\frac{3}{4} u^{\frac{4}{3}} + C - 6 \int u^{- \frac{2}{3}} d u$

Step 7

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

$\frac{3}{4} u^{\frac{4}{3}} + C - 6 (3 u^{\frac{1}{3}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{3}{4} u^{\frac{4}{3}} - 6 \cdot 3 u^{\frac{1}{3}} + C$

Step 8.2

Multiply $- 6$ by $3$ .

$\frac{3}{4} u^{\frac{4}{3}} - 18 u^{\frac{1}{3}} + C$

Step 9

Replace all occurrences of $u$ with $3 - y$ .

$\frac{3}{4} {(3 - y)}^{\frac{4}{3}} - 18 {(3 - y)}^{\frac{1}{3}} + C$