Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

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 $u = x - 1$ .

$u_{upper} = 2 - 1$

Step 1.5

Subtract $1$ from $2$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = - 1$

$u_{upper} = 1$

Step 1.7

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

$\int_{- 1}^{1} \frac{1}{u^{\frac{2}{3}}} d u$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Move $u^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(u^{\frac{2}{3}})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

$\int_{- 1}^{1} u^{\frac{2}{3} \cdot - 1} d u$

Step 2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

Tap for more steps...

Step 2.2.2.1

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

$\int_{- 1}^{1} u^{\frac{2 \cdot - 1}{3}} d u$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$\int_{- 1}^{1} u^{\frac{- 2}{3}} d u$

Step 2.2.3

Move the negative in front of the fraction.

$\int_{- 1}^{1} u^{- \frac{2}{3}} d u$

Step 3

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

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

Step 4

Substitute and simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

One to any power is one.

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

Step 4.2.2

Multiply $3$ by $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Rewrite the expression.

$3 - 3 {(- 1)}^{1}$

Step 4.2.6

Evaluate the exponent.

$3 - 3 \cdot - 1$

Step 4.2.7

Multiply $- 3$ by $- 1$ .

$3 + 3$

Step 4.2.8

Add $3$ and $3$ .

$6$

Step 5