Calculus Examples

$\int_{- 1}^{0} \sqrt{x - 1} d x$

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

Substitute the lower limit in for $x$ in $u = x - 1$ .

$u_{lower} = - 1 - 1$

Step 1.3

Subtract $1$ from $- 1$ .

$u_{lower} = - 2$

Step 1.4

Substitute the upper limit in for $x$ in $u = x - 1$ .

$u_{upper} = 0 - 1$

Step 1.5

Subtract $1$ from $0$ .

$u_{upper} = - 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 2$

$u_{upper} = - 1$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{- 2}^{- 1} \sqrt{u} d u$

Step 2

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

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

Step 3

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

$\frac{2}{3} u^{\frac{3}{2}}]_{- 2}^{- 1}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $- 1$ and at $- 2$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Combine $\frac{2}{3}$ and ${(- 1)}^{\frac{3}{2}}$ .

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

Step 4.2.2

Combine ${(- 2)}^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$\frac{2 {(- 1)}^{\frac{3}{2}}}{3} - \frac{{(- 2)}^{\frac{3}{2}} \cdot 2}{3}$

Step 4.2.3

Move $2$ to the left of ${(- 2)}^{\frac{3}{2}}$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite ${(- 1)}^{\frac{3}{2}}$ as $\sqrt{{(- 1)}^{3}}$ .

$\frac{2 \sqrt{{(- 1)}^{3}}}{3} - \frac{2 {(- 2)}^{\frac{3}{2}}}{3}$

Step 5.1.2

Raise $- 1$ to the power of $3$ .

$\frac{2 \sqrt{- 1}}{3} - \frac{2 {(- 2)}^{\frac{3}{2}}}{3}$

Step 5.1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{2 i}{3} - \frac{2 {(- 2)}^{\frac{3}{2}}}{3}$

Step 5.2

Reorder $\frac{2 i}{3}$ and $- \frac{2 {(- 2)}^{\frac{3}{2}}}{3}$ .

$- \frac{2 {(- 2)}^{\frac{3}{2}}}{3} + \frac{2 i}{3}$

Step 6