Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

Expand $(2 - \frac{1}{p^{2}}) (2 - \frac{1}{p^{2}})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2$ by $2$ .

$\int 4 + 2 (- \frac{1}{p^{2}}) - \frac{1}{p^{2}} \cdot 2 - \frac{1}{p^{2}} (- \frac{1}{p^{2}}) d p$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Multiply $- 1$ by $2$ .

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

Step 1.3.1.2.2

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

$\int 4 + \frac{- 2}{p^{2}} - \frac{1}{p^{2}} \cdot 2 - \frac{1}{p^{2}} (- \frac{1}{p^{2}}) d p$

Step 1.3.1.3

Move the negative in front of the fraction.

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

Step 1.3.1.4

Multiply $- \frac{1}{p^{2}} \cdot 2$ .

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Step 1.3.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.3.1.4.2

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

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

Step 1.3.1.5

Move the negative in front of the fraction.

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

Step 1.3.1.6

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

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$\int 4 - \frac{2}{p^{2}} - \frac{2}{p^{2}} + 1 \frac{1}{p^{2}} \frac{1}{p^{2}} d p$

Step 1.3.1.6.2

Multiply $\frac{1}{p^{2}}$ by $1$ .

$\int 4 - \frac{2}{p^{2}} - \frac{2}{p^{2}} + \frac{1}{p^{2}} \cdot \frac{1}{p^{2}} d p$

Step 1.3.1.6.3

Multiply $\frac{1}{p^{2}}$ by $\frac{1}{p^{2}}$ .

$\int 4 - \frac{2}{p^{2}} - \frac{2}{p^{2}} + \frac{1}{p^{2} p^{2}} d p$

Step 1.3.1.6.4

Multiply $p^{2}$ by $p^{2}$ by adding the exponents.

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Step 1.3.1.6.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 4 - \frac{2}{p^{2}} - \frac{2}{p^{2}} + \frac{1}{p^{2 + 2}} d p$

Step 1.3.1.6.4.2

Add $2$ and $2$ .

$\int 4 - \frac{2}{p^{2}} - \frac{2}{p^{2}} + \frac{1}{p^{4}} d p$

Step 1.3.2

Subtract $\frac{2}{p^{2}}$ from $- \frac{2}{p^{2}}$ .

$\int 4 - 2 \frac{2}{p^{2}} + \frac{1}{p^{4}} d p$

Step 1.4

Simplify each term.

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Step 1.4.1

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

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Step 1.4.1.1

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

$\int 4 + \frac{- 2 \cdot 2}{p^{2}} + \frac{1}{p^{4}} d p$

Step 1.4.1.2

Multiply $- 2$ by $2$ .

$\int 4 + \frac{- 4}{p^{2}} + \frac{1}{p^{4}} d p$

Step 1.4.2

Move the negative in front of the fraction.

$\int 4 - \frac{4}{p^{2}} + \frac{1}{p^{4}} d p$

Step 2

Split the single integral into multiple integrals.

$\int 4 d p + \int - \frac{4}{p^{2}} d p + \int \frac{1}{p^{4}} d p$

Step 3

Apply the constant rule.

$4 p + C + \int - \frac{4}{p^{2}} d p + \int \frac{1}{p^{4}} d p$

Step 4

Since $- 1$ is constant with respect to $p$ , move $- 1$ out of the integral.

$4 p + C - \int \frac{4}{p^{2}} d p + \int \frac{1}{p^{4}} d p$

Step 5

Since $4$ is constant with respect to $p$ , move $4$ out of the integral.

$4 p + C - (4 \int \frac{1}{p^{2}} d p) + \int \frac{1}{p^{4}} d p$

Step 6

Simplify the expression.

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Step 6.1

Multiply $4$ by $- 1$ .

$4 p + C - 4 \int \frac{1}{p^{2}} d p + \int \frac{1}{p^{4}} d p$

Step 6.2

Move $p^{2}$ out of the denominator by raising it to the $- 1$ power.

$4 p + C - 4 \int {(p^{2})}^{- 1} d p + \int \frac{1}{p^{4}} d p$

Step 6.3

Multiply the exponents in ${(p^{2})}^{- 1}$ .

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Step 6.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 p + C - 4 \int p^{2 \cdot - 1} d p + \int \frac{1}{p^{4}} d p$

Step 6.3.2

Multiply $2$ by $- 1$ .

$4 p + C - 4 \int p^{- 2} d p + \int \frac{1}{p^{4}} d p$

Step 7

By the Power Rule, the integral of $p^{- 2}$ with respect to $p$ is $- p^{- 1}$ .

$4 p + C - 4 (- p^{- 1} + C) + \int \frac{1}{p^{4}} d p$

Step 8

Apply basic rules of exponents.

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Step 8.1

Move $p^{4}$ out of the denominator by raising it to the $- 1$ power.

$4 p + C - 4 (- p^{- 1} + C) + \int {(p^{4})}^{- 1} d p$

Step 8.2

Multiply the exponents in ${(p^{4})}^{- 1}$ .

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Step 8.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 p + C - 4 (- p^{- 1} + C) + \int p^{4 \cdot - 1} d p$

Step 8.2.2

Multiply $4$ by $- 1$ .

$4 p + C - 4 (- p^{- 1} + C) + \int p^{- 4} d p$

Step 9

By the Power Rule, the integral of $p^{- 4}$ with respect to $p$ is $- \frac{1}{3} p^{- 3}$ .

$4 p + C - 4 (- p^{- 1} + C) - \frac{1}{3} p^{- 3} + C$

Step 10

Simplify.

$4 p + \frac{4}{p} - \frac{1}{3} p^{- 3} + C$