Calculus Examples

$\int_{0}^{1} \frac{1}{{(x - 1)}^{\frac{1}{3}}} d x$

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 $u = x - 1$ .

$u_{lower} = 0 - 1$

Step 1.3

Subtract $1$ from $0$ .

$u_{lower} = - 1$

Step 1.4

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

$u_{upper} = 1 - 1$

Step 1.5

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = - 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{- 1}^{0} \frac{1}{u^{\frac{1}{3}}} d u$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{- 1}^{0} {(u^{\frac{1}{3}})}^{- 1} d u$

Step 2.2

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 2.2.1

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

$\int_{- 1}^{0} u^{\frac{1}{3} \cdot - 1} d u$

Step 2.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\int_{- 1}^{0} u^{\frac{- 1}{3}} d u$

Step 2.2.3

Move the negative in front of the fraction.

$\int_{- 1}^{0} u^{- \frac{1}{3}} d u$

Step 3

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

$\frac{3}{2} u^{\frac{2}{3}}]_{- 1}^{0}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{3}{2} u^{\frac{2}{3}}$ at $0$ and at $- 1$ .

$(\frac{3}{2} \cdot 0^{\frac{2}{3}}) - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2

Simplify.

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Step 4.2.1

Rewrite $0$ as $0^{3}$ .

$\frac{3}{2} \cdot {(0^{3})}^{\frac{2}{3}} - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.2

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

$\frac{3}{2} \cdot 0^{3 (\frac{2}{3})} - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.3

Cancel the common factor of $3$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{3}{2} \cdot 0^{3 (\frac{2}{3})} - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.3.2

Rewrite the expression.

$\frac{3}{2} \cdot 0^{2} - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.4

Raising $0$ to any positive power yields $0$ .

$\frac{3}{2} \cdot 0 - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.5

Multiply $\frac{3}{2}$ by $0$ .

$0 - \frac{3}{2} {(- 1)}^{\frac{2}{3}}$

Step 4.2.6

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$0 - \frac{3}{2} {({(- 1)}^{3})}^{\frac{2}{3}}$

Step 4.2.7

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

$0 - \frac{3}{2} {(- 1)}^{3 (\frac{2}{3})}$

Step 4.2.8

Cancel the common factor of $3$ .

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Step 4.2.8.1

Cancel the common factor.

$0 - \frac{3}{2} {(- 1)}^{3 (\frac{2}{3})}$

Step 4.2.8.2

Rewrite the expression.

$0 - \frac{3}{2} {(- 1)}^{2}$

Step 4.2.9

Raise $- 1$ to the power of $2$ .

$0 - \frac{3}{2} \cdot 1$

Step 4.2.10

Multiply $- 1$ by $1$ .

$0 - \frac{3}{2}$

Step 4.2.11

Subtract $\frac{3}{2}$ from $0$ .

$- \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2}$

Decimal Form:

$- 1.5$

Mixed Number Form:

$- 1 \frac{1}{2}$

Step 6