Calculus Examples

$\int_{0}^{1} 11 x^{10} {(1 + x^{11})}^{10} d x$

Step 1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int_{0}^{1} x^{10} {(1 + x^{11})}^{10} d x$

Step 2

Let $u = 1 + x^{11}$ . Then $d u = 11 x^{10} d x$ , so $\frac{1}{11} d u = x^{10} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = 1 + x^{11}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $1 + x^{11}$ .

$\frac{d}{d x} [1 + x^{11}]$

Step 2.1.2

By the Sum Rule, the derivative of $1 + x^{11}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{11}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{11}]$

Step 2.1.3

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

$0 + \frac{d}{d x} [x^{11}]$

Step 2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 11$ .

$0 + 11 x^{10}$

Step 2.1.5

Add $0$ and $11 x^{10}$ .

$11 x^{10}$

Step 2.2

Substitute the lower limit in for $x$ in $u = 1 + x^{11}$ .

$u_{lower} = 1 + 0^{11}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

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

$u_{lower} = 1 + 0$

Step 2.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = 1 + x^{11}$ .

$u_{upper} = 1 + 1^{11}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 2.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 2.7

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

$11 \int_{1}^{2} u^{10} \frac{1}{11} d u$

Step 3

Combine $u^{10}$ and $\frac{1}{11}$ .

$11 \int_{1}^{2} \frac{u^{10}}{11} d u$

Step 4

Since $\frac{1}{11}$ is constant with respect to $u$ , move $\frac{1}{11}$ out of the integral.

$11 (\frac{1}{11} \int_{1}^{2} u^{10} d u)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine $\frac{1}{11}$ and $11$ .

$\frac{11}{11} \int_{1}^{2} u^{10} d u$

Step 5.2

Cancel the common factor of $11$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{11}{11} \int_{1}^{2} u^{10} d u$

Step 5.2.2

Rewrite the expression.

$1 \int_{1}^{2} u^{10} d u$

Step 5.3

Multiply $\int_{1}^{2} u^{10} d u$ by $1$ .

$\int_{1}^{2} u^{10} d u$

Step 6

By the Power Rule, the integral of $u^{10}$ with respect to $u$ is $\frac{1}{11} u^{11}$ .

$\frac{1}{11} u^{11}]_{1}^{2}$

Step 7

Substitute and simplify.

Tap for more steps...

Step 7.1

Evaluate $\frac{1}{11} u^{11}$ at $2$ and at $1$ .

$(\frac{1}{11} \cdot 2^{11}) - \frac{1}{11} \cdot 1^{11}$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Raise $2$ to the power of $11$ .

$\frac{1}{11} \cdot 2048 - \frac{1}{11} \cdot 1^{11}$

Step 7.2.2

Combine $\frac{1}{11}$ and $2048$ .

$\frac{2048}{11} - \frac{1}{11} \cdot 1^{11}$

Step 7.2.3

One to any power is one.

$\frac{2048}{11} - \frac{1}{11} \cdot 1$

Step 7.2.4

Multiply $- 1$ by $1$ .

$\frac{2048}{11} - \frac{1}{11}$

Step 7.2.5

Combine the numerators over the common denominator.

$\frac{2048 - 1}{11}$

Step 7.2.6

Subtract $1$ from $2048$ .

$\frac{2047}{11}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2047}{11}$

Decimal Form:

$186.\overline{09}$

Mixed Number Form:

$186 \frac{1}{11}$

Step 9