Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$\ln (| x |)]_{1}^{4} + \int_{1}^{4} - \frac{1}{x^{2}} d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\ln (| x |)]_{1}^{4} - \int_{1}^{4} \frac{1}{x^{2}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\ln (| x |)]_{1}^{4} - \int_{1}^{4} {(x^{2})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

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

$\ln (| x |)]_{1}^{4} - \int_{1}^{4} x^{2 \cdot - 1} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$\ln (| x |)]_{1}^{4} - \int_{1}^{4} x^{- 2} d x$

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

Evaluate $\ln (| x |)$ at $4$ and at $1$ .

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

Step 6.1.2

Evaluate $- x^{- 1}$ at $4$ and at $1$ .

$\ln (| 4 |) - \ln (| 1 |) - (- 4^{- 1} + 1^{- 1})$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\ln (| 4 |) - \ln (| 1 |) - (- \frac{1}{4} + 1^{- 1})$

Step 6.1.3.2

One to any power is one.

$\ln (| 4 |) - \ln (| 1 |) - (- \frac{1}{4} + 1)$

Step 6.1.3.3

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

$\ln (| 4 |) - \ln (| 1 |) - (- \frac{1}{4} + \frac{4}{4})$

Step 6.1.3.4

Combine the numerators over the common denominator.

$\ln (| 4 |) - \ln (| 1 |) - \frac{- 1 + 4}{4}$

Step 6.1.3.5

Add $- 1$ and $4$ .

$\ln (| 4 |) - \ln (| 1 |) - \frac{3}{4}$

Step 6.2

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

$\ln (\frac{| 4 |}{| 1 |}) - \frac{3}{4}$

Step 6.3

Simplify.

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Step 6.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\ln (\frac{4}{| 1 |}) - \frac{3}{4}$

Step 6.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\ln (\frac{4}{1}) - \frac{3}{4}$

Step 6.3.3

Divide $4$ by $1$ .

$\ln (4) - \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\ln (4) - \frac{3}{4}$

Decimal Form:

$0.63629436 \dots$

Step 8