Calculus Examples

$\int \frac{4 x - 2}{{(2 x - 1)}^{2}} d x$

Step 1

Let $u = 2 x - 1$ . 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 1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 1$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

$\int \frac{u}{u^{2}} d u$

Step 2

Cancel the common factor of $u$ and $u^{2}$ .

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Step 2.1

Raise $u$ to the power of $1$ .

$\int \frac{u^{1}}{u^{2}} d u$

Step 2.2

Factor $u$ out of $u^{1}$ .

$\int \frac{u \cdot 1}{u^{2}} d u$

Step 2.3

Cancel the common factors.

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Step 2.3.1

Factor $u$ out of $u^{2}$ .

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

Rewrite the expression.

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

Step 3

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

$\ln (| u |) + C$

Step 4

Replace all occurrences of $u$ with $2 x - 1$ .

$\ln (| 2 x - 1 |) + C$