Calculus Examples

$\int_{- 1}^{0} 2 x d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{- 1}^{0} x d x$

Step 2

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

$2 (\frac{1}{2} x^{2}]_{- 1}^{0})$

Step 3

Simplify the answer.

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Step 3.1

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

$2 (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 1$ .

$2 ((\frac{0^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raising $0$ to any positive power yields $0$ .

$2 (\frac{0}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.2

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

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Step 3.2.2.2.1

Factor $2$ out of $0$ .

$2 (\frac{2 (0)}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $2$ out of $2$ .

$2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.2.2.2

Cancel the common factor.

$2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.2.2.3

Rewrite the expression.

$2 (\frac{0}{1} - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.2.2.4

Divide $0$ by $1$ .

$2 (0 - \frac{{(- 1)}^{2}}{2})$

Step 3.2.2.3

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

$2 (0 - \frac{1}{2})$

Step 3.2.2.4

Subtract $\frac{1}{2}$ from $0$ .

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

Step 3.2.2.5

Multiply $- 1$ by $2$ .

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

Step 3.2.2.6

Combine $- 2$ and $\frac{1}{2}$ .

$\frac{- 2}{2}$

Step 3.2.2.7

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

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Step 3.2.2.7.1

Factor $2$ out of $- 2$ .

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

Step 3.2.2.7.2

Cancel the common factors.

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Step 3.2.2.7.2.1

Factor $2$ out of $2$ .

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

Step 3.2.2.7.2.2

Cancel the common factor.

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

Step 3.2.2.7.2.3

Rewrite the expression.

$\frac{- 1}{1}$

Step 3.2.2.7.2.4

Divide $- 1$ by $1$ .

$- 1$

Step 4

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