Calculus Examples

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

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 $u = x - 1$ .

$u_{lower} = 1 - 1$

Step 1.3

Subtract $1$ from $1$ .

$u_{lower} = 0$

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.

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

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

$\int_{0}^{1} (u + 1) \sqrt{u} d u$

Step 2

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

$\int_{0}^{1} (u + 1) u^{\frac{1}{2}} d u$

Step 3

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

Tap for more steps...

Step 3.1

Apply the distributive property.

$\int_{0}^{1} u \cdot u^{\frac{1}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.2

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

$\int_{0}^{1} u^{1} u^{\frac{1}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.3

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

$\int_{0}^{1} u^{1 + \frac{1}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.4

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

$\int_{0}^{1} u^{\frac{2}{2} + \frac{1}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.5

Combine the numerators over the common denominator.

$\int_{0}^{1} u^{\frac{2 + 1}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.6

Add $2$ and $1$ .

$\int_{0}^{1} u^{\frac{3}{2}} + 1 u^{\frac{1}{2}} d u$

Step 3.7

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

$\int_{0}^{1} u^{\frac{3}{2}} + u^{\frac{1}{2}} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{0}^{1} u^{\frac{3}{2}} d u + \int_{0}^{1} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{2}{5} u^{\frac{5}{2}}]_{0}^{1} + \int_{0}^{1} u^{\frac{1}{2}} d u$

Step 6

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

$\frac{2}{5} u^{\frac{5}{2}}]_{0}^{1} + \frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}$

Step 7

Combine $\frac{2}{5} u^{\frac{5}{2}}]_{0}^{1}$ and $\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}$ .

$\frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}$

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Evaluate $\frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}}$ at $1$ and at $0$ .

$(\frac{2}{5} \cdot 1^{\frac{5}{2}} + \frac{2}{3} \cdot 1^{\frac{3}{2}}) - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 8.2

One to any power is one.

$\frac{2}{5} \cdot 1 + \frac{2}{3} \cdot 1^{\frac{3}{2}} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 8.3

Multiply $\frac{2}{5}$ by $1$ .

$\frac{2}{5} + \frac{2}{3} \cdot 1^{\frac{3}{2}} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 8.4

One to any power is one.

$\frac{2}{5} + \frac{2}{3} \cdot 1 - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 8.5

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{5} + \frac{2}{3} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9

Simplify.

Tap for more steps...

Step 9.1

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{5} \cdot \frac{3}{3} + \frac{2}{3} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.2

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2}{5} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{5}{5} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.3.1

Multiply $\frac{2}{5}$ by $\frac{3}{3}$ .

$\frac{2 \cdot 3}{5 \cdot 3} + \frac{2}{3} \cdot \frac{5}{5} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.3.2

Multiply $5$ by $3$ .

$\frac{2 \cdot 3}{15} + \frac{2}{3} \cdot \frac{5}{5} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.3.3

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$\frac{2 \cdot 3}{15} + \frac{2 \cdot 5}{3 \cdot 5} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.3.4

Multiply $3$ by $5$ .

$\frac{2 \cdot 3}{15} + \frac{2 \cdot 5}{15} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.4

Combine the numerators over the common denominator.

$\frac{2 \cdot 3 + 2 \cdot 5}{15} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.5

Simplify the numerator.

Tap for more steps...

Step 9.5.1

Multiply $2$ by $3$ .

$\frac{6 + 2 \cdot 5}{15} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.5.2

Multiply $2$ by $5$ .

$\frac{6 + 10}{15} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 9.5.3

Add $6$ and $10$ .

$\frac{16}{15} - (\frac{2}{5} \cdot 0^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Rewrite $0$ as $0^{2}$ .

$\frac{16}{15} - (\frac{2}{5} \cdot {(0^{2})}^{\frac{5}{2}} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 10.2

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

$\frac{16}{15} - (\frac{2}{5} \cdot 0^{2 (\frac{5}{2})} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 10.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.3.1

Cancel the common factor.

$\frac{16}{15} - (\frac{2}{5} \cdot 0^{2 (\frac{5}{2})} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 10.3.2

Rewrite the expression.

$\frac{16}{15} - (\frac{2}{5} \cdot 0^{5} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 10.4

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

$\frac{16}{15} - (\frac{2}{5} \cdot 0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 11

Multiply $\frac{2}{5}$ by $0$ .

$\frac{16}{15} - (0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Rewrite $0$ as $0^{2}$ .

$\frac{16}{15} - (0 + \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 12.2

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

$\frac{16}{15} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 12.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.3.1

Cancel the common factor.

$\frac{16}{15} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 12.3.2

Rewrite the expression.

$\frac{16}{15} - (0 + \frac{2}{3} \cdot 0^{3})$

Step 12.4

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

$\frac{16}{15} - (0 + \frac{2}{3} \cdot 0)$

Step 13

Simplify the expression.

Tap for more steps...

Step 13.1

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

$\frac{16}{15} - (0 + 0)$

Step 13.2

Add $0$ and $0$ .

$\frac{16}{15} - 0$

Step 13.3

Multiply $- 1$ by $0$ .

$\frac{16}{15} + 0$

Step 13.4

Add $\frac{16}{15}$ and $0$ .

$\frac{16}{15}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{15}$

Decimal Form:

$1.0\overline{6}$

Mixed Number Form:

$1 \frac{1}{15}$