Calculus Examples

$\int_{0}^{5} (f (x) + g (x)) d x$

Step 1

Remove parentheses.

$\int_{0}^{5} f (x) + g (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{5} f (x) d x + \int_{0}^{5} g x d x$

Step 3

Apply the constant rule.

$f (x) x]_{0}^{5} + \int_{0}^{5} g x d x$

Step 4

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

$f (x) x]_{0}^{5} + g \int_{0}^{5} x d x$

Step 5

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

$f (x) x]_{0}^{5} + g \frac{1}{2} x^{2}]_{0}^{5}$

Step 6

Simplify the answer.

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Step 6.1

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

$f (x) x]_{0}^{5} + g \frac{x^{2}}{2}]_{0}^{5}$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $f (x) x$ at $5$ and at $0$ .

$(f (x) \cdot 5) - f (x) \cdot 0 + g \frac{x^{2}}{2}]_{0}^{5}$

Step 6.2.2

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $0$ .

$f (x) \cdot 5 - f (x) \cdot 0 + g (\frac{5^{2}}{2} - \frac{0^{2}}{2})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Move $5$ to the left of $f (x)$ .

$5 \cdot f (x) - f (x) \cdot 0 + g (\frac{5^{2}}{2} - \frac{0^{2}}{2})$

Step 6.2.3.2

Multiply $0$ by $- 1$ .

$5 f (x) + 0 f (x) + g (\frac{5^{2}}{2} - \frac{0^{2}}{2})$

Step 6.2.3.3

Multiply $0$ by $f (x)$ .

$5 f (x) + 0 + g (\frac{5^{2}}{2} - \frac{0^{2}}{2})$

Step 6.2.3.4

Add $5 f (x)$ and $0$ .

$5 f (x) + g (\frac{5^{2}}{2} - \frac{0^{2}}{2})$

Step 6.2.3.5

Raise $5$ to the power of $2$ .

$5 f (x) + g (\frac{25}{2} - \frac{0^{2}}{2})$

Step 6.2.3.6

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

$5 f (x) + g (\frac{25}{2} - \frac{0}{2})$

Step 6.2.3.7

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

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Step 6.2.3.7.1

Factor $2$ out of $0$ .

$5 f (x) + g (\frac{25}{2} - \frac{2 (0)}{2})$

Step 6.2.3.7.2

Cancel the common factors.

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Step 6.2.3.7.2.1

Factor $2$ out of $2$ .

$5 f (x) + g (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 6.2.3.7.2.2

Cancel the common factor.

$5 f (x) + g (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 6.2.3.7.2.3

Rewrite the expression.

$5 f (x) + g (\frac{25}{2} - \frac{0}{1})$

Step 6.2.3.7.2.4

Divide $0$ by $1$ .

$5 f (x) + g (\frac{25}{2} - 0)$

Step 6.2.3.8

Multiply $- 1$ by $0$ .

$5 f (x) + g (\frac{25}{2} + 0)$

Step 6.2.3.9

Add $\frac{25}{2}$ and $0$ .

$5 f (x) + g \frac{25}{2}$

Step 6.2.3.10

Combine $g$ and $\frac{25}{2}$ .

$5 f (x) + \frac{g \cdot 25}{2}$

Step 6.2.3.11

Move $25$ to the left of $g$ .

$5 f (x) + \frac{25 g}{2}$

Step 6.3

Reorder terms.

$5 f (x) + \frac{25}{2} g$

Step 7

Combine $\frac{25}{2}$ and $g$ .

$5 f (x) + \frac{25 g}{2}$

Step 8