Calculus Examples

$\int_{1}^{2} \frac{6 x - 1}{3 x^{2} - x} d x$

Step 1

Let $u = 3 x^{2} - x$ . Then $d u = (6 x - 1) d x$ , so $\frac{1}{6 x - 1} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $3 x^{2} - x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.3.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- x]$

Step 1.1.4

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

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Step 1.1.4.1

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

$6 x - \frac{d}{d x} [x]$

Step 1.1.4.2

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

$6 x - 1 \cdot 1$

Step 1.1.4.3

Multiply $- 1$ by $1$ .

$6 x - 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x^{2} - x$ .

$u_{lower} = 3 \cdot 1^{2} - 1 \cdot 1$

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.

$u_{lower} = 3 \cdot 1 - 1 \cdot 1$

Step 1.3.1.2

Multiply $3$ by $1$ .

$u_{lower} = 3 - 1 \cdot 1$

Step 1.3.1.3

Multiply $- 1$ by $1$ .

$u_{lower} = 3 - 1$

Step 1.3.2

Subtract $1$ from $3$ .

$u_{lower} = 2$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x^{2} - x$ .

$u_{upper} = 3 \cdot 2^{2} - 1 \cdot 2$

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 $2$ to the power of $2$ .

$u_{upper} = 3 \cdot 4 - 1 \cdot 2$

Step 1.5.1.2

Multiply $3$ by $4$ .

$u_{upper} = 12 - 1 \cdot 2$

Step 1.5.1.3

Multiply $- 1$ by $2$ .

$u_{upper} = 12 - 2$

Step 1.5.2

Subtract $2$ from $12$ .

$u_{upper} = 10$

Step 1.6

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

$u_{lower} = 2$

$u_{upper} = 10$

Step 1.7

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

$\int_{2}^{10} \frac{1}{u} d u$

Step 2

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

$\ln (| u |)]_{2}^{10}$

Step 3

Evaluate $\ln (| u |)$ at $10$ and at $2$ .

$\ln (| 10 |) - \ln (| 2 |)$

Step 4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 10 |}{| 2 |})$

Step 5

Simplify.

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Step 5.1

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$\ln (\frac{10}{| 2 |})$

Step 5.2

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\ln (\frac{10}{2})$

Step 5.3

Divide $10$ by $2$ .

$\ln (5)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (5)$

Decimal Form:

$1.60943791 \dots$

Step 7