Calculus Examples

$\int_{1}^{4 x} \frac{9}{t} d t$

Step 1

Since

$9$ is constant with respect to

$t$ , move

$9$ out of the integral.

$9 \int_{1}^{4 x} \frac{1}{t} d t$

Step 2

The integral of

$\frac{1}{t}$ with respect to

$t$ is

$\ln (| t |)$ .

$9 (\ln (| t |)]_{1}^{4 x})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate

$\ln (| t |)$ at

$4 x$ and at

$1$ .

$9 (\ln (| 4 x |) - \ln (| 1 |))$

Step 3.2

Use the quotient property of logarithms,

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$9 \ln (\frac{| 4 x |}{| 1 |})$

Step 3.3

Reorder terms.

$9 \ln (\frac{1}{| 1 |} | 4 x |)$

Step 4

Simplify.

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Step 4.1

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

$0$ and

$1$ is

$1$ .

$9 \ln (\frac{1}{1} | 4 x |)$

Step 4.2

Divide

$1$ by

$1$ .

$9 \ln (1 | 4 x |)$

Step 4.3

Multiply

$| 4 x |$ by

$1$ .

$9 \ln (| 4 x |)$

Step 4.4

Remove non-negative terms from the absolute value.

$9 \ln (4 | x |)$