Calculus Examples

$\int_{p}^{2 p} x d x$

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

Factor $2$ out of $2 p$ .

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

Step 2.2.2

Apply the product rule to $2 (p)$ .

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

Step 2.2.3

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

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

Step 2.2.4

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} p^{2} - \frac{1}{2} p^{2}$

Step 2.2.5

Combine $\frac{4}{2}$ and $p^{2}$ .

$\frac{4 p^{2}}{2} - \frac{1}{2} p^{2}$

Step 2.2.6

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

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Step 2.2.6.1

Factor $2$ out of $4 p^{2}$ .

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

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

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

Step 2.2.6.2.2

Cancel the common factor.

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

Step 2.2.6.2.3

Rewrite the expression.

$\frac{2 p^{2}}{1} - \frac{1}{2} p^{2}$

Step 2.2.6.2.4

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

$2 p^{2} - \frac{1}{2} p^{2}$

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $p^{2}$ out of $2 p^{2} \cdot 2 - p^{2}$ .

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Step 3.5.1.1

Factor $p^{2}$ out of $2 p^{2} \cdot 2$ .

$\frac{p^{2} (2 \cdot 2) - p^{2}}{2}$

Step 3.5.1.2

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

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

Step 3.5.1.3

Factor $p^{2}$ out of $p^{2} (2 \cdot 2) + p^{2} \cdot - 1$ .

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

Step 3.5.2

Multiply $2$ by $2$ .

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

Step 3.5.3

Subtract $1$ from $4$ .

$\frac{p^{2} \cdot 3}{2}$

Step 3.6

Move $3$ to the left of $p^{2}$ .

$\frac{3 p^{2}}{2}$