Calculus Examples

$\int \frac{x + 2}{{(2 x - 4)}^{\frac{3}{2}}} d x$

Step 1

Let $u = 2 x - 4$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x - 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 4$ .

$\frac{d}{d x} [2 x - 4]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 4$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 4]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 4]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 4]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Add $2$ and $2$ .

$\int \frac{\frac{u}{2} + 4}{u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{\frac{u}{2} + 4}{u^{\frac{3}{2}}}$ by $\frac{2}{2}$ .

$\int \frac{2}{2} \cdot \frac{\frac{u}{2} + 4}{u^{\frac{3}{2}}} \frac{1}{2} d u$

Step 2.3

Combine.

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

Step 2.4

Apply the distributive property.

$\int \frac{2 \frac{u}{2} + 2 \cdot 4}{2 u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

$\int \frac{2 \frac{u}{2} + 2 \cdot 4}{2 u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.5.2

Rewrite the expression.

$\int \frac{u + 2 \cdot 4}{2 u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.6

Multiply $2$ by $4$ .

$\int \frac{u + 8}{2 u^{\frac{3}{2}}} \cdot \frac{1}{2} d u$

Step 2.7

Multiply $\frac{u + 8}{2 u^{\frac{3}{2}}}$ by $\frac{1}{2}$ .

$\int \frac{u + 8}{2 u^{\frac{3}{2}} \cdot 2} d u$

Step 2.8

Multiply $2$ by $2$ .

$\int \frac{u + 8}{4 u^{\frac{3}{2}}} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{u + 8}{u^{\frac{3}{2}}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{4} \int (u + 8) {(u^{\frac{3}{2}})}^{- 1} d u$

Step 4.2

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

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Step 4.2.1

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

$\frac{1}{4} \int (u + 8) u^{\frac{3}{2} \cdot - 1} d u$

Step 4.2.2

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

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Step 4.2.2.1

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

$\frac{1}{4} \int (u + 8) u^{\frac{3 \cdot - 1}{2}} d u$

Step 4.2.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{4} \int (u + 8) u^{\frac{- 3}{2}} d u$

Step 4.2.3

Move the negative in front of the fraction.

$\frac{1}{4} \int (u + 8) u^{- \frac{3}{2}} d u$

Step 5

Expand $(u + 8) u^{- \frac{3}{2}}$ .

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Step 5.1

Apply the distributive property.

$\frac{1}{4} \int u \cdot u^{- \frac{3}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.2

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

$\frac{1}{4} \int u^{1} u^{- \frac{3}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.3

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

$\frac{1}{4} \int u^{1 - \frac{3}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.4

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

$\frac{1}{4} \int u^{\frac{2}{2} - \frac{3}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.5

Combine the numerators over the common denominator.

$\frac{1}{4} \int u^{\frac{2 - 3}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.6

Subtract $3$ from $2$ .

$\frac{1}{4} \int u^{\frac{- 1}{2}} + 8 u^{- \frac{3}{2}} d u$

Step 5.7

Reorder $u^{\frac{- 1}{2}}$ and $8 u^{- \frac{3}{2}}$ .

$\frac{1}{4} \int 8 u^{- \frac{3}{2}} + u^{\frac{- 1}{2}} d u$

Step 6

Move the negative in front of the fraction.

$\frac{1}{4} \int 8 u^{- \frac{3}{2}} + u^{- \frac{1}{2}} d u$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int 8 u^{- \frac{3}{2}} d u + \int u^{- \frac{1}{2}} d u)$

Step 8

Since $8$ is constant with respect to $u$ , move $8$ out of the integral.

$\frac{1}{4} (8 \int u^{- \frac{3}{2}} d u + \int u^{- \frac{1}{2}} d u)$

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

$\frac{1}{4} (- \frac{16}{u^{\frac{1}{2}}} + 2 u^{\frac{1}{2}}) + C$

Step 12

Replace all occurrences of $u$ with $2 x - 4$ .

$\frac{1}{4} (- \frac{16}{{(2 x - 4)}^{\frac{1}{2}}} + 2 {(2 x - 4)}^{\frac{1}{2}}) + C$

Step 13

Simplify.

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Step 13.1

To write $2 {(2 x - 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(2 x - 4)}^{\frac{1}{2}}}{{(2 x - 4)}^{\frac{1}{2}}}$ .

$\frac{1}{4} (- \frac{16}{{(2 x - 4)}^{\frac{1}{2}}} + \frac{2 {(2 x - 4)}^{\frac{1}{2}} {(2 x - 4)}^{\frac{1}{2}}}{{(2 x - 4)}^{\frac{1}{2}}}) + C$

Step 13.2

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{- 16 + 2 {(2 x - 4)}^{\frac{1}{2}} {(2 x - 4)}^{\frac{1}{2}}}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Multiply ${(2 x - 4)}^{\frac{1}{2}}$ by ${(2 x - 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 13.3.1.1

Move ${(2 x - 4)}^{\frac{1}{2}}$ .

$\frac{1}{4} \cdot \frac{- 16 + 2 ({(2 x - 4)}^{\frac{1}{2}} {(2 x - 4)}^{\frac{1}{2}})}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.1.2

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

$\frac{1}{4} \cdot \frac{- 16 + 2 {(2 x - 4)}^{\frac{1}{2} + \frac{1}{2}}}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.1.3

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{- 16 + 2 {(2 x - 4)}^{\frac{1 + 1}{2}}}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.1.4

Add $1$ and $1$ .

$\frac{1}{4} \cdot \frac{- 16 + 2 {(2 x - 4)}^{\frac{2}{2}}}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.1.5

Divide $2$ by $2$ .

$\frac{1}{4} \cdot \frac{- 16 + 2 {(2 x - 4)}^{1}}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.2

Simplify $2 {(2 x - 4)}^{1}$ .

$\frac{1}{4} \cdot \frac{- 16 + 2 (2 x - 4)}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.3

Apply the distributive property.

$\frac{1}{4} \cdot \frac{- 16 + 2 (2 x) + 2 \cdot - 4}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.4

Multiply $2$ by $2$ .

$\frac{1}{4} \cdot \frac{- 16 + 4 x + 2 \cdot - 4}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.5

Multiply $2$ by $- 4$ .

$\frac{1}{4} \cdot \frac{- 16 + 4 x - 8}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.6

Subtract $8$ from $- 16$ .

$\frac{1}{4} \cdot \frac{4 x - 24}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.7

Factor $4$ out of $4 x - 24$ .

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Step 13.3.7.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} \cdot \frac{4 (x) - 24}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.7.2

Factor $4$ out of $- 24$ .

$\frac{1}{4} \cdot \frac{4 x + 4 \cdot - 6}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.3.7.3

Factor $4$ out of $4 x + 4 \cdot - 6$ .

$\frac{1}{4} \cdot \frac{4 (x - 6)}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.4

Combine.

$\frac{1 (4 (x - 6))}{4 {(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.5

Cancel the common factor.

$\frac{1 (4 (x - 6))}{4 {(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.6

Rewrite the expression.

$\frac{1 (x - 6)}{{(2 x - 4)}^{\frac{1}{2}}} + C$

Step 13.7

Multiply $x - 6$ by $1$ .

$\frac{x - 6}{{(2 x - 4)}^{\frac{1}{2}}} + C$