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.

$9 \ln (4 | x |)$