Calculus Examples

$\frac{2}{5 - 2 x} + \frac{2}{x} + \frac{3}{x^{2}}$

Step 1

Write $\frac{2}{5 - 2 x} + \frac{2}{x} + \frac{3}{x^{2}}$ as a function.

$f (x) = \frac{2}{5 - 2 x} + \frac{2}{x} + \frac{3}{x^{2}}$

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.

$F (x) = \int \frac{2}{5 - 2 x} + \frac{2}{x} + \frac{3}{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{2}{5 - 2 x} d x + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{1}{5 - 2 x} d x + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 6

Let $u = 5 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = 5 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $5 - 2 x$ .

$\frac{d}{d x} [5 - 2 x]$

Step 6.1.2

Differentiate.

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Step 6.1.2.1

By the Sum Rule, the derivative of $5 - 2 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x]$

Step 6.1.2.2

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

$0 + \frac{d}{d x} [- 2 x]$

Step 6.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 6.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 2 \cdot 1$

Step 6.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 6.1.4

Subtract $2$ from $0$ .

$- 2$

Step 6.2

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

$2 \int \frac{1}{u} \cdot \frac{1}{- 2} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$2 \int \frac{1}{u} (- \frac{1}{2}) d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 7.2

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$2 \int - \frac{1}{u \cdot 2} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 7.3

Move $2$ to the left of $u$ .

$2 \int - \frac{1}{2 u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 8

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

$2 (- \int \frac{1}{2 u} d u) + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 9

Multiply $- 1$ by $2$ .

$- 2 \int \frac{1}{2 u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 10

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

$- 2 (\frac{1}{2} \int \frac{1}{u} d u) + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11

Simplify.

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Step 11.1

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

$\frac{- 2}{2} \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11.2

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

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Step 11.2.1

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2} \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 1}{2 (1)} \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 1} \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11.2.2.3

Rewrite the expression.

$\frac{- 1}{1} \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 11.2.2.4

Divide $- 1$ by $1$ .

$- \int \frac{1}{u} d u + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 12

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- (\ln (| u |) + C) + \int \frac{2}{x} d x + \int \frac{3}{x^{2}} d x$

Step 13

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$- (\ln (| u |) + C) + 2 \int \frac{1}{x} d x + \int \frac{3}{x^{2}} d x$

Step 14

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + \int \frac{3}{x^{2}} d x$

Step 15

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

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + 3 \int \frac{1}{x^{2}} d x$

Step 16

Apply basic rules of exponents.

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Step 16.1

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

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + 3 \int {(x^{2})}^{- 1} d x$

Step 16.2

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

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Step 16.2.1

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

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + 3 \int x^{2 \cdot - 1} d x$

Step 16.2.2

Multiply $2$ by $- 1$ .

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + 3 \int x^{- 2} d x$

Step 17

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

$- (\ln (| u |) + C) + 2 (\ln (| x |) + C) + 3 (- x^{- 1} + C)$

Step 18

Simplify.

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Step 18.1

Simplify.

$- \ln (| u |) + 2 \ln (| x |) + 3 (- \frac{1}{x}) + C$

Step 18.2

Simplify.

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Step 18.2.1

Multiply $- 1$ by $3$ .

$- \ln (| u |) + 2 \ln (| x |) - 3 \frac{1}{x} + C$

Step 18.2.2

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

$- \ln (| u |) + 2 \ln (| x |) + \frac{- 3}{x} + C$

Step 18.2.3

Move the negative in front of the fraction.

$- \ln (| u |) + 2 \ln (| x |) - \frac{3}{x} + C$

Step 19

Replace all occurrences of $u$ with $5 - 2 x$ .

$- \ln (| 5 - 2 x |) + 2 \ln (| x |) - \frac{3}{x} + C$

Step 20

The answer is the antiderivative of the function $f (x) = \frac{2}{5 - 2 x} + \frac{2}{x} + \frac{3}{x^{2}}$ .

$F (x) =$ $- \ln (| 5 - 2 x |) + 2 \ln (| x |) - \frac{3}{x} + C$