Calculus Examples

$\int_{0}^{2} 2 f (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor of $4$ and $2$ .

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Step 3.2.2.2.1

Factor $2$ out of $4$ .

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

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 3.2.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.2.3

Rewrite the expression.

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

Step 3.2.2.2.2.4

Divide $2$ by $1$ .

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

Step 3.2.2.3

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

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

Step 3.2.2.4

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

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Step 3.2.2.4.1

Factor $2$ out of $0$ .

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

Step 3.2.2.4.2

Cancel the common factors.

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Step 3.2.2.4.2.1

Factor $2$ out of $2$ .

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

Step 3.2.2.4.2.2

Cancel the common factor.

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

Step 3.2.2.4.2.3

Rewrite the expression.

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

Step 3.2.2.4.2.4

Divide $0$ by $1$ .

$2 f (2 - 0)$

Step 3.2.2.5

Multiply $- 1$ by $0$ .

$2 f (2 + 0)$

Step 3.2.2.6

Add $2$ and $0$ .

$2 f \cdot 2$

Step 3.2.2.7

Multiply $2$ by $2$ .

$4 f$

Step 4