Calculus Examples

| x - 1 |

$| x - 1 |$

Step 1

Set the argument in the absolute value equal to

0

$0$ to find the potential values to split the solution at.

x - 1 = 0

$x - 1 = 0$

Step 2

Simplify the answer.

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Step 2.1

Solve the equation for

x

$x$ .

x = 1

$x = 1$

Step 2.2

Create intervals around the solutions to find where

x - 1

$x - 1$ is positive and negative.

(- \infty, 1), (1, \infty)

$(- \infty, 1), (1, \infty)$

Step 2.3

Substitute a value from each interval into

x - 1

$x - 1$ to figure out where the expression is positive or negative.

\begin{array}{c} Interval & Sign on interval \\ (- \infty, 1) & - \\ (1, \infty) & + \end{array}

$\begin{array}{c} Interval & Sign on interval \\ (- \infty, 1) & - \\ (1, \infty) & + \end{array}$

Step 2.4

Integrate the argument of the absolute value.

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Step 2.4.1

Set up the integral with the argument of the absolute value.

\int x - 1 d x

$\int x - 1 d x$

Step 2.4.2

Split the single integral into multiple integrals.

\int x d x + \int - 1 d x

$\int x d x + \int - 1 d x$

Step 2.4.3

By the Power Rule, the integral of

x

$x$ with respect to

x

$x$ is

\frac{1}{2} x^{2}

$\frac{1}{2} x^{2}$ .

\frac{1}{2} x^{2} + C + \int - 1 d x

$\frac{1}{2} x^{2} + C + \int - 1 d x$

Step 2.4.4

Apply the constant rule.

\frac{1}{2} x^{2} + C - x + C

$\frac{1}{2} x^{2} + C - x + C$

Step 2.4.5

Simplify.

\frac{1}{2} x^{2} - x + C

$\frac{1}{2} x^{2} - x + C$

\frac{1}{2} x^{2} - x + C

$\frac{1}{2} x^{2} - x + C$

Step 2.5

On the intervals where the argument is negative, multiply the solution of the integral by

- 1

$- 1$ .

{\begin{cases} - (\frac{1}{2} x^{2} - x + C) & x \leq 1 \\ \frac{1}{2} x^{2} - x + C & x > 1 \end{cases}

${\begin{cases} - (\frac{1}{2} x^{2} - x + C) & x \leq 1 \\ \frac{1}{2} x^{2} - x + C & x > 1 \end{cases}$

Step 2.6

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{2}

$x^{2}$ .

{\begin{cases} - (\frac{x^{2}}{2} - x + C) & x \leq 1 \\ \frac{1}{2} x^{2} - x + C & x > 1 \end{cases}

${\begin{cases} - (\frac{x^{2}}{2} - x + C) & x \leq 1 \\ \frac{1}{2} x^{2} - x + C & x > 1 \end{cases}$

Step 2.7

Simplify.

{\begin{cases} - \frac{x^{2}}{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C

${\begin{cases} - \frac{x^{2}}{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C$

Step 2.8

Simplify.

{\begin{cases} - \frac{1}{2} x^{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C

${\begin{cases} - \frac{1}{2} x^{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C$

{\begin{cases} - \frac{1}{2} x^{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C

${\begin{cases} - \frac{1}{2} x^{2} + x & x \leq 1 \\ \frac{1}{2} x^{2} - x & x > 1 \end{cases} + C$