Calculus Examples

\int \frac{1}{5 - 3 x} d x

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

Step 1

Let

u = 5 - 3 x

$u = 5 - 3 x$ . Then

d u = - 3 d x

$d u = - 3 d x$ , so

- \frac{1}{3} d u = d x

$- \frac{1}{3} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = 5 - 3 x

$u = 5 - 3 x$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

5 - 3 x

$5 - 3 x$ .

\frac{d}{d x} [5 - 3 x]

$\frac{d}{d x} [5 - 3 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of

5 - 3 x

$5 - 3 x$ with respect to

x

$x$ is

\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]

$\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]$ .

\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]

$\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]$

Step 1.1.2.2

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5

$5$ with respect to

x

$x$ is

0

$0$ .

0 + \frac{d}{d x} [- 3 x]

$0 + \frac{d}{d x} [- 3 x]$

0 + \frac{d}{d x} [- 3 x]

$0 + \frac{d}{d x} [- 3 x]$

Step 1.1.3

Evaluate

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

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

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Step 1.1.3.1

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3 x

$- 3 x$ with respect to

x

$x$ is

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

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

0 - 3 \frac{d}{d x} [x]

$0 - 3 \frac{d}{d x} [x]$

Step 1.1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

0 - 3 \cdot 1

$0 - 3 \cdot 1$

Step 1.1.3.3

Multiply

- 3

$- 3$ by

1

$1$ .

0 - 3

$0 - 3$

0 - 3

$0 - 3$

Step 1.1.4

Subtract

3

$3$ from

0

$0$ .

- 3

$- 3$

- 3

$- 3$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int \frac{1}{u} \cdot \frac{1}{- 3} d u

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

\int \frac{1}{u} \cdot \frac{1}{- 3} d u

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

\int \frac{1}{u} (- \frac{1}{3}) d u

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

Step 2.2

Multiply

\frac{1}{u}

$\frac{1}{u}$ by

\frac{1}{3}

$\frac{1}{3}$ .

\int - \frac{1}{u \cdot 3} d u

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

Step 2.3

Move

3

$3$ to the left of

u

$u$ .

\int - \frac{1}{3 u} d u

$\int - \frac{1}{3 u} d u$

\int - \frac{1}{3 u} d u

$\int - \frac{1}{3 u} d u$

Step 3

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

- \int \frac{1}{3 u} d u

$- \int \frac{1}{3 u} d u$

Step 4

Since

\frac{1}{3}

$\frac{1}{3}$ is constant with respect to

$u$ , move

$\frac{1}{3}$ out of the integral.

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

Step 5

The integral of

$\frac{1}{u}$ with respect to

$u$ is

$\ln (| u |)$ .

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

Step 6

Simplify.

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

Step 7

Replace all occurrences of

$u$ with

$5 - 3 x$ .

$- \frac{1}{3} \ln (| 5 - 3 x |) + C$

$\int_{}^{} \frac{d x}{5 - 3 x}$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$