Calculus Examples

$f (x) = \frac{1}{\sqrt{2 x - 1}}$

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.

$F (x) = \int \frac{1}{\sqrt{2 x - 1}} d x$

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $2 x - 1$ .

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

Step 3.1.2

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 3.1.3

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

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Step 3.1.3.1

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 3.1.3.2

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 3.1.3.3

Multiply $2$ by $1$ .

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

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

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

$2 + 0$

Step 3.1.4.2

Add $2$ and $0$ .

$2$

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Move $2$ to the left of $\sqrt{u}$ .

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

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 6.2

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

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

Step 6.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

Move the negative in front of the fraction.

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

Step 7

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{1}{2} (2 u^{\frac{1}{2}} + C)$ as $\frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C$ .

$\frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{2}{2} u^{\frac{1}{2}} + C$

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{2}{2} u^{\frac{1}{2}} + C$

Step 8.2.2.2

Rewrite the expression.

$1 u^{\frac{1}{2}} + C$

Step 8.2.3

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

$u^{\frac{1}{2}} + C$

Step 9

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

${(2 x - 1)}^{\frac{1}{2}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{1}{\sqrt{2 x - 1}}$ .

$F (x) =$ ${(2 x - 1)}^{\frac{1}{2}} + C$