Evaluate the Integral

$\int_{- 1}^{0} \frac{4 x}{5} d x$

Step 1

Since $\frac{4}{5}$ is constant with respect to $x$ , move $\frac{4}{5}$ out of the integral.

$\frac{4}{5} \int_{- 1}^{0} x d x$

Step 2

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

$\frac{4}{5} \frac{1}{2} x^{2}]_{- 1}^{0}$

Step 3

Substitute and simplify.

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Step 3.1

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

$\frac{4}{5} ((\frac{1}{2} \cdot 0^{2}) - \frac{1}{2} {(- 1)}^{2})$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{4}{5} (\frac{1}{2} \cdot 0 - \frac{1}{2} {(- 1)}^{2})$

Step 3.2.2

Multiply $\frac{1}{2}$ by $0$ .

$\frac{4}{5} (0 - \frac{1}{2} {(- 1)}^{2})$

Step 3.2.3

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

$\frac{4}{5} (0 - \frac{1}{2} \cdot 1)$

Step 3.2.4

Multiply $- 1$ by $1$ .

$\frac{4}{5} (0 - \frac{1}{2})$

Step 3.2.5

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

$\frac{4}{5} (- \frac{1}{2})$

Step 3.2.6

Multiply $\frac{4}{5}$ by $\frac{1}{2}$ .

$- \frac{4}{5 \cdot 2}$

Step 3.2.7

Multiply $5$ by $2$ .

$- \frac{4}{10}$

Step 3.2.8

Cancel the common factor of $4$ and $10$ .

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Step 3.2.8.1

Factor $2$ out of $4$ .

$- \frac{2 (2)}{10}$

Step 3.2.8.2

Cancel the common factors.

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Step 3.2.8.2.1

Factor $2$ out of $10$ .

$- \frac{2 \cdot 2}{2 \cdot 5}$

Step 3.2.8.2.2

Cancel the common factor.

$- \frac{2 \cdot 2}{2 \cdot 5}$

Step 3.2.8.2.3

Rewrite the expression.

$- \frac{2}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{5}$

Decimal Form:

$- 0.4$

Step 5

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Evaluate the Integral

Integrals

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