Calculus Examples

$f (x) = \frac{3}{{(5 - 7 x)}^{4}}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \frac{3}{{(5 - 7 x)}^{4}} d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{1}{{(5 - 7 x)}^{4}} d x$

Step 4

Let $u = 5 - 7 x$ . Then $d u = - 7 d x$ , so $- \frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 5 - 7 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $5 - 7 x$ .

$\frac{d}{d x} [5 - 7 x]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $5 - 7 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 7 x]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- 7 x]$

Step 4.1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 7 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 4.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$0 - 7 \frac{d}{d x} [x]$

Step 4.1.3.2

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

$0 - 7 \cdot 1$

Step 4.1.3.3

Multiply $- 7$ by $1$ .

$0 - 7$

Step 4.1.4

Subtract $7$ from $0$ .

$- 7$

Step 4.2

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

$3 \int \frac{1}{u^{4}} \cdot \frac{1}{- 7} d u$

Step 5

Simplify.

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Step 5.1

Move the negative in front of the fraction.

$3 \int \frac{1}{u^{4}} (- \frac{1}{7}) d u$

Step 5.2

Multiply $\frac{1}{u^{4}}$ by $\frac{1}{7}$ .

$3 \int - \frac{1}{u^{4} \cdot 7} d u$

Step 5.3

Move $7$ to the left of $u^{4}$ .

$3 \int - \frac{1}{7 u^{4}} d u$

Step 6

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

$3 (- \int \frac{1}{7 u^{4}} d u)$

Step 7

Multiply $- 1$ by $3$ .

$- 3 \int \frac{1}{7 u^{4}} d u$

Step 8

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$- 3 (\frac{1}{7} \int \frac{1}{u^{4}} d u)$

Step 9

Simplify the expression.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{7}$ and $- 3$ .

$\frac{- 3}{7} \int \frac{1}{u^{4}} d u$

Step 9.1.2

Move the negative in front of the fraction.

$- \frac{3}{7} \int \frac{1}{u^{4}} d u$

Step 9.2

Apply basic rules of exponents.

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Step 9.2.1

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

$- \frac{3}{7} \int {(u^{4})}^{- 1} d u$

Step 9.2.2

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

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Step 9.2.2.1

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

$- \frac{3}{7} \int u^{4 \cdot - 1} d u$

Step 9.2.2.2

Multiply $4$ by $- 1$ .

$- \frac{3}{7} \int u^{- 4} d u$

Step 10

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

$- \frac{3}{7} (- \frac{1}{3} u^{- 3} + C)$

Step 11

Simplify.

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Step 11.1

Rewrite $- \frac{3}{7} (- \frac{1}{3} u^{- 3} + C)$ as $- \frac{3}{7} (- \frac{1}{3} \cdot \frac{1}{u^{3}}) + C$ .

$- \frac{3}{7} (- \frac{1}{3} \cdot \frac{1}{u^{3}}) + C$

Step 11.2

Simplify.

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Step 11.2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{3}$ .

$- \frac{3}{7} (- \frac{1}{u^{3} \cdot 3}) + C$

Step 11.2.2

Move $3$ to the left of $u^{3}$ .

$- \frac{3}{7} (- \frac{1}{3 \cdot u^{3}}) + C$

Step 11.2.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{3}{7}) \frac{1}{3 u^{3}} + C$

Step 11.2.4

Multiply $\frac{3}{7}$ by $1$ .

$\frac{3}{7} \cdot \frac{1}{3 u^{3}} + C$

Step 11.2.5

Multiply $\frac{3}{7}$ by $\frac{1}{3 u^{3}}$ .

$\frac{3}{7 (3 u^{3})} + C$

Step 11.2.6

Multiply $3$ by $7$ .

$\frac{3}{21 u^{3}} + C$

Step 11.2.7

Cancel the common factor of $3$ and $21$ .

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Step 11.2.7.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{21 u^{3}} + C$

Step 11.2.7.2

Cancel the common factors.

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Step 11.2.7.2.1

Factor $3$ out of $21 u^{3}$ .

$\frac{3 (1)}{3 (7 u^{3})} + C$

Step 11.2.7.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 (7 u^{3})} + C$

Step 11.2.7.2.3

Rewrite the expression.

$\frac{1}{7 u^{3}} + C$

Step 12

Replace all occurrences of $u$ with $5 - 7 x$ .

$\frac{1}{7 {(5 - 7 x)}^{3}} + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{3}{{(5 - 7 x)}^{4}}$ .

$F (x) =$ $\frac{1}{7 {(5 - 7 x)}^{3}} + C$