Calculus Examples

$\int_{4 p}^{7 p} x d x$

Step 1

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

$\frac{1}{2} x^{2}]_{4 p}^{7 p}$

Step 2

Simplify the answer.

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Step 2.1

Substitute and simplify.

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Step 2.1.1

Evaluate $\frac{1}{2} x^{2}$ at $7 p$ and at $4 p$ .

$(\frac{1}{2} {(7 p)}^{2}) - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Factor $7$ out of $7 p$ .

$\frac{1}{2} {(7 (p))}^{2} - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2.2

Apply the product rule to $7 (p)$ .

$\frac{1}{2} (7^{2} p^{2}) - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2.3

Raise $7$ to the power of $2$ .

$\frac{1}{2} (49 p^{2}) - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2.4

Combine $49$ and $\frac{1}{2}$ .

$\frac{49}{2} p^{2} - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2.5

Combine $\frac{49}{2}$ and $p^{2}$ .

$\frac{49 p^{2}}{2} - \frac{1}{2} {(4 p)}^{2}$

Step 2.1.2.6

Factor $4$ out of $4 p$ .

$\frac{49 p^{2}}{2} - \frac{1}{2} {(4 (p))}^{2}$

Step 2.1.2.7

Apply the product rule to $4 (p)$ .

$\frac{49 p^{2}}{2} - \frac{1}{2} (4^{2} p^{2})$

Step 2.1.2.8

Raise $4$ to the power of $2$ .

$\frac{49 p^{2}}{2} - \frac{1}{2} (16 p^{2})$

Step 2.1.2.9

Multiply $16$ by $- 1$ .

$\frac{49 p^{2}}{2} - 16 (\frac{1}{2}) p^{2}$

Step 2.1.2.10

Combine $- 16$ and $\frac{1}{2}$ .

$\frac{49 p^{2}}{2} + \frac{- 16}{2} p^{2}$

Step 2.1.2.11

Combine $\frac{- 16}{2}$ and $p^{2}$ .

$\frac{49 p^{2}}{2} + \frac{- 16 p^{2}}{2}$

Step 2.1.2.12

Cancel the common factor of $- 16$ and $2$ .

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Step 2.1.2.12.1

Factor $2$ out of $- 16 p^{2}$ .

$\frac{49 p^{2}}{2} + \frac{2 (- 8 p^{2})}{2}$

Step 2.1.2.12.2

Cancel the common factors.

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Step 2.1.2.12.2.1

Factor $2$ out of $2$ .

$\frac{49 p^{2}}{2} + \frac{2 (- 8 p^{2})}{2 (1)}$

Step 2.1.2.12.2.2

Cancel the common factor.

$\frac{49 p^{2}}{2} + \frac{2 (- 8 p^{2})}{2 \cdot 1}$

Step 2.1.2.12.2.3

Rewrite the expression.

$\frac{49 p^{2}}{2} + \frac{- 8 p^{2}}{1}$

Step 2.1.2.12.2.4

Divide $- 8 p^{2}$ by $1$ .

$\frac{49 p^{2}}{2} - 8 p^{2}$

Step 2.1.2.13

To write $- 8 p^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{49 p^{2}}{2} - 8 p^{2} \cdot \frac{2}{2}$

Step 2.1.2.14

Combine $- 8 p^{2}$ and $\frac{2}{2}$ .

$\frac{49 p^{2}}{2} + \frac{- 8 p^{2} \cdot 2}{2}$

Step 2.1.2.15

Combine the numerators over the common denominator.

$\frac{49 p^{2} - 8 p^{2} \cdot 2}{2}$

Step 2.1.2.16

Multiply $2$ by $- 8$ .

$\frac{49 p^{2} - 16 p^{2}}{2}$

Step 2.1.2.17

Subtract $16 p^{2}$ from $49 p^{2}$ .

$\frac{33 p^{2}}{2}$

Step 2.2

Reorder terms.

$\frac{33}{2} p^{2}$

Step 3

Combine $\frac{33}{2}$ and $p^{2}$ .

$\frac{33 p^{2}}{2}$