Calculus Examples

$\int \frac{x}{\sqrt{x - 1}} d x$

Step 1

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

Expand $(u + 1) u^{- \frac{1}{2}}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4

Write $1$ as a fraction with a common denominator.

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Subtract $1$ from $2$ .

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

Step 3.7

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

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

Step 6

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

$\frac{2}{3} u^{\frac{3}{2}} + C + 2 u^{\frac{1}{2}} + C$

Step 7

Simplify.

$\frac{2}{3} u^{\frac{3}{2}} + 2 u^{\frac{1}{2}} + C$

Step 8

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

$\frac{2}{3} {(x - 1)}^{\frac{3}{2}} + 2 {(x - 1)}^{\frac{1}{2}} + C$