Calculus Examples

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

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

One to any power is one.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $0$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 4