Calculus Examples

$\int_{2}^{5} (3 f (x) + 2) d x$

Step 1

Remove parentheses.

$\int_{2}^{5} 3 f (x) + 2 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{5} 3 f x d x + \int_{2}^{5} 2 d x$

Step 3

Since $3 f$ is constant with respect to $x$ , move $3 f$ out of the integral.

$3 f \int_{2}^{5} x d x + \int_{2}^{5} 2 d x$

Step 4

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

$3 f \frac{1}{2} x^{2}]_{2}^{5} + \int_{2}^{5} 2 d x$

Step 5

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

$3 f \frac{x^{2}}{2}]_{2}^{5} + \int_{2}^{5} 2 d x$

Step 6

Apply the constant rule.

$3 f \frac{x^{2}}{2}]_{2}^{5} + 2 x]_{2}^{5}$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

$3 f ((\frac{5^{2}}{2}) - \frac{2^{2}}{2}) + 2 x]_{2}^{5}$

Step 7.1.2

Evaluate $2 x$ at $5$ and at $2$ .

$3 f (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3

Simplify.

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Step 7.1.3.1

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

$3 f (\frac{25}{2} - \frac{2^{2}}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.2

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

$3 f (\frac{25}{2} - \frac{4}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.3

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

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Step 7.1.3.3.1

Factor $2$ out of $4$ .

$3 f (\frac{25}{2} - \frac{2 \cdot 2}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.3.2

Cancel the common factors.

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Step 7.1.3.3.2.1

Factor $2$ out of $2$ .

$3 f (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.3.2.2

Cancel the common factor.

$3 f (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.3.2.3

Rewrite the expression.

$3 f (\frac{25}{2} - \frac{2}{1}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.3.2.4

Divide $2$ by $1$ .

$3 f (\frac{25}{2} - 1 \cdot 2) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.4

Multiply $- 1$ by $2$ .

$3 f (\frac{25}{2} - 2) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.5

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 f (\frac{25}{2} - 2 \cdot \frac{2}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.6

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

$3 f (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.7

Combine the numerators over the common denominator.

$3 f \frac{25 - 2 \cdot 2}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.8

Simplify the numerator.

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Step 7.1.3.8.1

Multiply $- 2$ by $2$ .

$3 f \frac{25 - 4}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.8.2

Subtract $4$ from $25$ .

$3 f \frac{21}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.9

Combine $3$ and $\frac{21}{2}$ .

$f \frac{3 \cdot 21}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.10

Multiply $3$ by $21$ .

$f \frac{63}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.11

Combine $f$ and $\frac{63}{2}$ .

$\frac{f \cdot 63}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.12

Move $63$ to the left of $f$ .

$\frac{63 \cdot f}{2} + 2 \cdot 5 - 2 \cdot 2$

Step 7.1.3.13

Multiply $2$ by $5$ .

$\frac{63 f}{2} + 10 - 2 \cdot 2$

Step 7.1.3.14

Multiply $- 2$ by $2$ .

$\frac{63 f}{2} + 10 - 4$

Step 7.1.3.15

Subtract $4$ from $10$ .

$\frac{63 f}{2} + 6$

Step 7.2

Reorder terms.

$\frac{63}{2} f + 6$

Step 8

Combine $\frac{63}{2}$ and $f$ .

$\frac{63 f}{2} + 6$

Step 9