Calculus Examples

$\int_{1}^{3} \frac{x - 2}{x^{2} - 4 x} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} - 4 x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$2 x - 4 \cdot 1$

Step 1.1.3.3

Multiply $- 4$ by $1$ .

$2 x - 4$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} - 4 x$ .

$u_{lower} = 1^{2} - 4 \cdot 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

One to any power is one.

$u_{lower} = 1 - 4 \cdot 1$

Step 1.3.1.2

Multiply $- 4$ by $1$ .

$u_{lower} = 1 - 4$

Step 1.3.2

Subtract $4$ from $1$ .

$u_{lower} = - 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} - 4 x$ .

$u_{upper} = 3^{2} - 4 \cdot 3$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $3$ to the power of $2$ .

$u_{upper} = 9 - 4 \cdot 3$

Step 1.5.1.2

Multiply $- 4$ by $3$ .

$u_{upper} = 9 - 12$

Step 1.5.2

Subtract $12$ from $9$ .

$u_{upper} = - 3$

Step 1.6

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

$u_{lower} = - 3$

$u_{upper} = - 3$

Step 1.7

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

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

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

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

Step 2.2

Move $2$ to the left of $u$ .

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

Step 3

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

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

Step 4

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

$\frac{1}{2} \ln (| u |)]_{- 3}^{- 3}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\ln (| u |)$ at $- 3$ and at $- 3$ .

$\frac{1}{2} ((\ln (| - 3 |)) - \ln (| - 3 |))$

Step 5.2

Simplify.

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Step 5.2.1

Subtract $\ln (| - 3 |)$ from $\ln (| - 3 |)$ .

$\frac{1}{2} \cdot 0$

Step 5.2.2

Multiply $\frac{1}{2}$ by $0$ .

$0$

Step 6