Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Evaluate $- 2 x$ at $1$ and at $0$ .

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

Step 6.3

Simplify.

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Step 6.3.1

One to any power is one.

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

Step 6.3.2

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

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

Step 6.3.3

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

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Step 6.3.3.1

Factor $2$ out of $0$ .

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

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $2$ .

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.3.2.4

Divide $0$ by $1$ .

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

Step 6.3.4

Multiply $- 1$ by $0$ .

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

Step 6.3.5

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

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

Step 6.3.6

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

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

Step 6.3.7

Cancel the common factor of $2$ .

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Step 6.3.7.1

Cancel the common factor.

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

Step 6.3.7.2

Rewrite the expression.

$1 - 2 \cdot 1 + 2 \cdot 0$

Step 6.3.8

Multiply $- 2$ by $1$ .

$1 - 2 + 2 \cdot 0$

Step 6.3.9

Multiply $2$ by $0$ .

$1 - 2 + 0$

Step 6.3.10

Add $- 2$ and $0$ .

$1 - 2$

Step 6.3.11

Subtract $2$ from $1$ .

$- 1$

Step 7

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