Calculus Examples

$f (x) = \frac{3}{2 x - 5} - 4$

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.

$F (x) = \int \frac{3}{2 x - 5} - 4 d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{3}{2 x - 5} d x + \int - 4 d x$

Step 4

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

$3 \int \frac{1}{2 x - 5} d x + \int - 4 d x$

Step 5

Let $u = 2 x - 5$ . 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 5.1

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

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Step 5.1.1

Differentiate $2 x - 5$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [- 5]$

Step 5.1.3.3

Multiply $2$ by $1$ .

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

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

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

$2 + 0$

Step 5.1.4.2

Add $2$ and $0$ .

$2$

Step 5.2

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

$3 \int \frac{1}{u} \cdot \frac{1}{2} d u + \int - 4 d x$

Step 6

Simplify.

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Step 6.1

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

$3 \int \frac{1}{u \cdot 2} d u + \int - 4 d x$

Step 6.2

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

$3 \int \frac{1}{2 u} d u + \int - 4 d x$

Step 7

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

$3 (\frac{1}{2} \int \frac{1}{u} d u) + \int - 4 d x$

Step 8

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

$\frac{3}{2} \int \frac{1}{u} d u + \int - 4 d x$

Step 9

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

$\frac{3}{2} (\ln (| u |) + C) + \int - 4 d x$

Step 10

Apply the constant rule.

$\frac{3}{2} (\ln (| u |) + C) - 4 x + C$

Step 11

Simplify.

$\frac{3}{2} \ln (| u |) - 4 x + C$

Step 12

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

$\frac{3}{2} \ln (| 2 x - 5 |) - 4 x + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{3}{2 x - 5} - 4$ .

$F (x) =$ $\frac{3}{2} \ln (| 2 x - 5 |) - 4 x + C$