Calculus Examples

$f (p) = 4 {(p^{2} - 1)}^{3} (2 p)$

Step 1

The function $F (p)$ can be found by finding the indefinite integral of the derivative $f (p)$ .

$F (p) = \int f (p) d p$

Step 2

Set up the integral to solve.

$F (p) = \int 4 {(p^{2} - 1)}^{3} (2 p) d p$

Step 3

Multiply $2$ by $4$ .

$\int 8 {(p^{2} - 1)}^{3} p d p$

Step 4

Since $8$ is constant with respect to $p$ , move $8$ out of the integral.

$8 \int {(p^{2} - 1)}^{3} p d p$

Step 5

Let $u = p^{2} - 1$ . Then $d u = 2 p d p$ , so $\frac{1}{2} d u = p d p$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = p^{2} - 1$ . Find $\frac{d u}{d p}$ .

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Step 5.1.1

Differentiate $p^{2} - 1$ .

$\frac{d}{d p} [p^{2} - 1]$

Step 5.1.2

By the Sum Rule, the derivative of $p^{2} - 1$ with respect to $p$ is $\frac{d}{d p} [p^{2}] + \frac{d}{d p} [- 1]$ .

$\frac{d}{d p} [p^{2}] + \frac{d}{d p} [- 1]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 2$ .

$2 p + \frac{d}{d p} [- 1]$

Step 5.1.4

Since $- 1$ is constant with respect to $p$ , the derivative of $- 1$ with respect to $p$ is $0$ .

$2 p + 0$

Step 5.1.5

Add $2 p$ and $0$ .

$2 p$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$8 \int u^{3} \frac{1}{2} d u$

Step 6

Combine $u^{3}$ and $\frac{1}{2}$ .

$8 \int \frac{u^{3}}{2} d u$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$8 (\frac{1}{2} \int u^{3} d u)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $8$ .

$\frac{8}{2} \int u^{3} d u$

Step 8.2

Cancel the common factor of $8$ and $2$ .

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Step 8.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} \int u^{3} d u$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1)} \int u^{3} d u$

Step 8.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} \int u^{3} d u$

Step 8.2.2.3

Rewrite the expression.

$\frac{4}{1} \int u^{3} d u$

Step 8.2.2.4

Divide $4$ by $1$ .

$4 \int u^{3} d u$

Step 9

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

$4 (\frac{1}{4} u^{4} + C)$

Step 10

Simplify.

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Step 10.1

Rewrite $4 (\frac{1}{4} u^{4} + C)$ as $4 (\frac{1}{4}) u^{4} + C$ .

$4 (\frac{1}{4}) u^{4} + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$\frac{4}{4} u^{4} + C$

Step 10.2.2

Cancel the common factor of $4$ .

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Step 10.2.2.1

Cancel the common factor.

$\frac{4}{4} u^{4} + C$

Step 10.2.2.2

Rewrite the expression.

$1 u^{4} + C$

Step 10.2.3

Multiply $u^{4}$ by $1$ .

$u^{4} + C$

Step 11

Replace all occurrences of $u$ with $p^{2} - 1$ .

${(p^{2} - 1)}^{4} + C$

Step 12

The answer is the antiderivative of the function $f (p) = 4 {(p^{2} - 1)}^{3} (2 p)$ .

$F (p) =$ ${(p^{2} - 1)}^{4} + C$