Calculus Examples

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

Step 1

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

$\frac{1}{2} x^{2}]_{- 1}^{1}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} x^{2}$ at $1$ and at $- 1$ .

$(\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} {(- 1)}^{2}$

Step 2.2

Simplify.

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Step 2.2.1

One to any power is one.

$\frac{1}{2} \cdot 1 - \frac{1}{2} {(- 1)}^{2}$

Step 2.2.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} - \frac{1}{2} {(- 1)}^{2}$

Step 2.2.3

Raise $- 1$ to the power of $2$ .

$\frac{1}{2} - \frac{1}{2} \cdot 1$

Step 2.2.4

Multiply $- 1$ by $1$ .

$\frac{1}{2} - \frac{1}{2}$

Step 2.2.5

Combine the numerators over the common denominator.

$\frac{1 - 1}{2}$

Step 2.2.6

Subtract $1$ from $1$ .

$\frac{0}{2}$

Step 2.2.7

Cancel the common factor of $0$ and $2$ .

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Step 2.2.7.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2}$

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 0}{2 \cdot 1}$

Step 2.2.7.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 \cdot 1}$

Step 2.2.7.2.3

Rewrite the expression.

$\frac{0}{1}$

Step 2.2.7.2.4

Divide $0$ by $1$ .

$0$

Step 3