Calculus Examples

$5 {(4 x - 3)}^{4} (4)$

Step 1

Write $5 {(4 x - 3)}^{4} (4)$ as a function.

$f (x) = 5 {(4 x - 3)}^{4} (4)$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int 5 {(4 x - 3)}^{4} (4) d x$

Step 4

Multiply $4$ by $5$ .

$\int 20 {(4 x - 3)}^{4} d x$

Step 5

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int {(4 x - 3)}^{4} d x$

Step 6

Let $u = 4 x - 3$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = 4 x - 3$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $4 x - 3$ .

$\frac{d}{d x} [4 x - 3]$

Step 6.1.2

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$

Step 6.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 6.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 6.1.3.2

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

$4 \cdot 1 + \frac{d}{d x} [- 3]$

Step 6.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 3]$

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$4 + 0$

Step 6.1.4.2

Add $4$ and $0$ .

$4$

Step 6.2

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

$20 \int u^{4} \frac{1}{4} d u$

Step 7

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

$20 \int \frac{u^{4}}{4} d u$

Step 8

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$20 (\frac{1}{4} \int u^{4} d u)$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{4}$ and $20$ .

$\frac{20}{4} \int u^{4} d u$

Step 9.2

Cancel the common factor of $20$ and $4$ .

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Step 9.2.1

Factor $4$ out of $20$ .

$\frac{4 \cdot 5}{4} \int u^{4} d u$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 5}{4 (1)} \int u^{4} d u$

Step 9.2.2.2

Cancel the common factor.

$\frac{4 \cdot 5}{4 \cdot 1} \int u^{4} d u$

Step 9.2.2.3

Rewrite the expression.

$\frac{5}{1} \int u^{4} d u$

Step 9.2.2.4

Divide $5$ by $1$ .

$5 \int u^{4} d u$

Step 10

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$5 (\frac{1}{5} u^{5} + C)$

Step 11

Simplify.

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Step 11.1

Rewrite $5 (\frac{1}{5} u^{5} + C)$ as $5 (\frac{1}{5}) u^{5} + C$ .

$5 (\frac{1}{5}) u^{5} + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} u^{5} + C$

Step 11.2.2

Cancel the common factor of $5$ .

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Step 11.2.2.1

Cancel the common factor.

$\frac{5}{5} u^{5} + C$

Step 11.2.2.2

Rewrite the expression.

$1 u^{5} + C$

Step 11.2.3

Multiply $u^{5}$ by $1$ .

$u^{5} + C$

Step 12

Replace all occurrences of $u$ with $4 x - 3$ .

${(4 x - 3)}^{5} + C$

Step 13

The answer is the antiderivative of the function $f (x) = 5 {(4 x - 3)}^{4} (4)$ .

$F (x) =$ ${(4 x - 3)}^{5} + C$