Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

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

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

Step 3.2.2.3

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

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Step 3.2.2.3.1

Factor $2$ out of $0$ .

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

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor $2$ out of $2$ .

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

Step 3.2.2.3.2.2

Cancel the common factor.

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

Step 3.2.2.3.2.3

Rewrite the expression.

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

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

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

Step 3.2.2.4

Multiply $- 1$ by $0$ .

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

Step 3.2.2.5

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

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

Step 3.2.2.6

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

$\frac{2}{2}$

Step 3.2.2.7

Cancel the common factor of $2$ .

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Step 3.2.2.7.1

Cancel the common factor.

$\frac{2}{2}$

Step 3.2.2.7.2

Rewrite the expression.

$1$

Step 4