Calculus Examples

$\int_{0}^{1} \frac{1}{1 - x} d x$

Step 1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 0$

Step 1.3

Subtract $0$ from $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 1.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{1}^{0} \frac{- 1}{u} d u$

Step 2

Split the fraction into multiple fractions.

$\int_{1}^{0} - \frac{1}{u} d u$

Step 3

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

$- \int_{1}^{0} \frac{1}{u} d u$

Step 4

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

$- (\ln (| u |)]_{1}^{0})$

Step 5

Evaluate $\ln (| u |)$ at $0$ and at $1$ .

$- (\ln (| 0 |) - \ln (| 1 |))$

Step 6

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

$- \ln (\frac{| 0 |}{| 1 |})$

Step 7

Simplify.

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Step 7.1

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

$- \ln (\frac{0}{| 1 |})$

Step 7.2

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

$- \ln (\frac{0}{1})$

Step 7.3

Divide $0$ by $1$ .

$- \ln (0)$

Step 7.4

The natural logarithm of zero is undefined.

Undefined

$- \ln (0)$

Step 8

The natural logarithm of zero is undefined.

Undefined