Calculus Examples

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

Step 1

Let $u = \frac{3}{2} y - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{3}{2} y - x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{3}{2} y - x$ .

$\frac{d}{d x} [\frac{3}{2} y - x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $\frac{3}{2} y - x$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{2} y] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [\frac{3}{2} y] + \frac{d}{d x} [- x]$

Step 1.1.2.2

Since $\frac{3}{2} y$ is constant with respect to $x$ , the derivative of $\frac{3}{2} y$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Multiply $\frac{- u + \frac{3 y}{2}}{u^{2}}$ by $\frac{2}{2}$ .

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

Step 2.3

Combine.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

Multiply $- 1$ by $2$ .

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

Step 3

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

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

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Subtract $2$ from $1$ .

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

Step 6.5

Reorder $- 2 u^{- 1}$ and $3 y u^{- 2}$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

The integral of $u^{- 1}$ with respect to $u$ is $\ln (| u |)$ .

$- \frac{1}{2} (3 y (- u^{- 1} + C) - 2 (\ln (| u |) + C))$

Step 12

Simplify.

$- \frac{1}{2} (- \frac{3 y}{u} - 2 \ln (| u |)) + C$

Step 13

Replace all occurrences of $u$ with $\frac{3}{2} y - x$ .

$- \frac{1}{2} (- \frac{3 y}{\frac{3}{2} y - x} - 2 \ln (| \frac{3}{2} y - x |)) + C$