Calculus Examples

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

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 $d u$ .

$\int (u + 1) \sqrt{u} d u$

Step 2

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

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

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

Add $2$ and $1$ .

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

Step 3.7

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$\frac{2}{5} u^{\frac{5}{2}} + C + \int u^{\frac{1}{2}} d u$

Step 6

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

$\frac{2}{5} u^{\frac{5}{2}} + C + \frac{2}{3} u^{\frac{3}{2}} + C$

Step 7

Simplify.

$\frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}} + C$

Step 8

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

$\frac{2}{5} {(x - 1)}^{\frac{5}{2}} + \frac{2}{3} {(x - 1)}^{\frac{3}{2}} + C$