Calculus Examples

$\int_{0}^{1} x^{11} {(1 + x^{12})}^{11} d x$

Step 1

Let $u = 1 + x^{12}$ . Then $d u = 12 x^{11} d x$ , so $\frac{1}{12} d u = x^{11} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 + x^{12}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 + x^{12}$ .

$\frac{d}{d x} [1 + x^{12}]$

Step 1.1.2

By the Sum Rule, the derivative of $1 + x^{12}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{12}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{12}]$

Step 1.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{12}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 12$ .

$0 + 12 x^{11}$

Step 1.1.5

Add $0$ and $12 x^{11}$ .

$12 x^{11}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + x^{12}$ .

$u_{lower} = 1 + 0^{12}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

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

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + x^{12}$ .

$u_{upper} = 1 + 1^{12}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

$\int_{1}^{2} u^{11} \frac{1}{12} d u$

Step 2

Combine $u^{11}$ and $\frac{1}{12}$ .

$\int_{1}^{2} \frac{u^{11}}{12} d u$

Step 3

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$\frac{1}{12} \int_{1}^{2} u^{11} d u$

Step 4

By the Power Rule, the integral of $u^{11}$ with respect to $u$ is $\frac{1}{12} u^{12}$ .

$\frac{1}{12} \frac{1}{12} u^{12}]_{1}^{2}$

Step 5

Substitute and simplify.

Tap for more steps...

Step 5.1

Evaluate $\frac{1}{12} u^{12}$ at $2$ and at $1$ .

$\frac{1}{12} ((\frac{1}{12} \cdot 2^{12}) - \frac{1}{12} \cdot 1^{12})$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Raise $2$ to the power of $12$ .

$\frac{1}{12} (\frac{1}{12} \cdot 4096 - \frac{1}{12} \cdot 1^{12})$

Step 5.2.2

Combine $\frac{1}{12}$ and $4096$ .

$\frac{1}{12} (\frac{4096}{12} - \frac{1}{12} \cdot 1^{12})$

Step 5.2.3

Cancel the common factor of $4096$ and $12$ .

Tap for more steps...

Step 5.2.3.1

Factor $4$ out of $4096$ .

$\frac{1}{12} (\frac{4 (1024)}{12} - \frac{1}{12} \cdot 1^{12})$

Step 5.2.3.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.2.1

Factor $4$ out of $12$ .

$\frac{1}{12} (\frac{4 \cdot 1024}{4 \cdot 3} - \frac{1}{12} \cdot 1^{12})$

Step 5.2.3.2.2

Cancel the common factor.

$\frac{1}{12} (\frac{4 \cdot 1024}{4 \cdot 3} - \frac{1}{12} \cdot 1^{12})$

Step 5.2.3.2.3

Rewrite the expression.

$\frac{1}{12} (\frac{1024}{3} - \frac{1}{12} \cdot 1^{12})$

Step 5.2.4

One to any power is one.

$\frac{1}{12} (\frac{1024}{3} - \frac{1}{12} \cdot 1)$

Step 5.2.5

Multiply $- 1$ by $1$ .

$\frac{1}{12} (\frac{1024}{3} - \frac{1}{12})$

Step 5.2.6

To write $\frac{1024}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{12} (\frac{1024}{3} \cdot \frac{4}{4} - \frac{1}{12})$

Step 5.2.7

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.2.7.1

Multiply $\frac{1024}{3}$ by $\frac{4}{4}$ .

$\frac{1}{12} (\frac{1024 \cdot 4}{3 \cdot 4} - \frac{1}{12})$

Step 5.2.7.2

Multiply $3$ by $4$ .

$\frac{1}{12} (\frac{1024 \cdot 4}{12} - \frac{1}{12})$

Step 5.2.8

Combine the numerators over the common denominator.

$\frac{1}{12} \cdot \frac{1024 \cdot 4 - 1}{12}$

Step 5.2.9

Simplify the numerator.

Tap for more steps...

Step 5.2.9.1

Multiply $1024$ by $4$ .

$\frac{1}{12} \cdot \frac{4096 - 1}{12}$

Step 5.2.9.2

Subtract $1$ from $4096$ .

$\frac{1}{12} \cdot \frac{4095}{12}$

Step 5.2.10

Cancel the common factor of $4095$ and $12$ .

Tap for more steps...

Step 5.2.10.1

Factor $3$ out of $4095$ .

$\frac{1}{12} \cdot \frac{3 (1365)}{12}$

Step 5.2.10.2

Cancel the common factors.

Tap for more steps...

Step 5.2.10.2.1

Factor $3$ out of $12$ .

$\frac{1}{12} \cdot \frac{3 \cdot 1365}{3 \cdot 4}$

Step 5.2.10.2.2

Cancel the common factor.

$\frac{1}{12} \cdot \frac{3 \cdot 1365}{3 \cdot 4}$

Step 5.2.10.2.3

Rewrite the expression.

$\frac{1}{12} \cdot \frac{1365}{4}$

Step 5.2.11

Multiply $\frac{1}{12}$ by $\frac{1365}{4}$ .

$\frac{1365}{12 \cdot 4}$

Step 5.2.12

Multiply $12$ by $4$ .

$\frac{1365}{48}$

Step 5.2.13

Cancel the common factor of $1365$ and $48$ .

Tap for more steps...

Step 5.2.13.1

Factor $3$ out of $1365$ .

$\frac{3 (455)}{48}$

Step 5.2.13.2

Cancel the common factors.

Tap for more steps...

Step 5.2.13.2.1

Factor $3$ out of $48$ .

$\frac{3 \cdot 455}{3 \cdot 16}$

Step 5.2.13.2.2

Cancel the common factor.

$\frac{3 \cdot 455}{3 \cdot 16}$

Step 5.2.13.2.3

Rewrite the expression.

$\frac{455}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{455}{16}$

Decimal Form:

$28.4375$

Mixed Number Form:

$28 \frac{7}{16}$

Step 7