Calculus Examples

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

Step 1

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

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

Step 2

Simplify the answer.

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Step 2.1

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

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

Step 2.2

Substitute and simplify.

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Step 2.2.1

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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Step 2.2.2.2.1

Factor $2$ out of $0$ .

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

Step 2.2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.2.4

Divide $0$ by $1$ .

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

Step 2.2.2.3

One to any power is one.

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

Step 2.2.2.4

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

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

Step 2.2.2.5

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.6

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

$\frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 4