Calculus Examples

$\int_{0}^{1} x^{9} {(1 + x^{10})}^{9} d x$

Step 1

Let $u = 1 + x^{10}$ . Then $d u = 10 x^{9} d x$ , so $\frac{1}{10} d u = x^{9} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 + x^{10}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + x^{10}$ .

$\frac{d}{d x} [1 + x^{10}]$

Step 1.1.2

By the Sum Rule, the derivative of $1 + x^{10}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{10}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{10}]$

Step 1.1.3

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

$0 + \frac{d}{d x} [x^{10}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$0 + 10 x^{9}$

Step 1.1.5

Add $0$ and $10 x^{9}$ .

$10 x^{9}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + x^{10}$ .

$u_{lower} = 1 + 0^{10}$

Step 1.3

Simplify.

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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 $u = 1 + x^{10}$ .

$u_{upper} = 1 + 1^{10}$

Step 1.5

Simplify.

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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.

$\int_{1}^{2} u^{9} \frac{1}{10} d u$

Step 2

Combine $u^{9}$ and $\frac{1}{10}$ .

$\int_{1}^{2} \frac{u^{9}}{10} d u$

Step 3

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$\frac{1}{10} \int_{1}^{2} u^{9} d u$

Step 4

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$\frac{1}{10} \frac{1}{10} u^{10}]_{1}^{2}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{10} u^{10}$ at $2$ and at $1$ .

$\frac{1}{10} ((\frac{1}{10} \cdot 2^{10}) - \frac{1}{10} \cdot 1^{10})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $2$ to the power of $10$ .

$\frac{1}{10} (\frac{1}{10} \cdot 1024 - \frac{1}{10} \cdot 1^{10})$

Step 5.2.2

Combine $\frac{1}{10}$ and $1024$ .

$\frac{1}{10} (\frac{1024}{10} - \frac{1}{10} \cdot 1^{10})$

Step 5.2.3

Cancel the common factor of $1024$ and $10$ .

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Step 5.2.3.1

Factor $2$ out of $1024$ .

$\frac{1}{10} (\frac{2 (512)}{10} - \frac{1}{10} \cdot 1^{10})$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $2$ out of $10$ .

$\frac{1}{10} (\frac{2 \cdot 512}{2 \cdot 5} - \frac{1}{10} \cdot 1^{10})$

Step 5.2.3.2.2

Cancel the common factor.

$\frac{1}{10} (\frac{2 \cdot 512}{2 \cdot 5} - \frac{1}{10} \cdot 1^{10})$

Step 5.2.3.2.3

Rewrite the expression.

$\frac{1}{10} (\frac{512}{5} - \frac{1}{10} \cdot 1^{10})$

Step 5.2.4

One to any power is one.

$\frac{1}{10} (\frac{512}{5} - \frac{1}{10} \cdot 1)$

Step 5.2.5

Multiply $- 1$ by $1$ .

$\frac{1}{10} (\frac{512}{5} - \frac{1}{10})$

Step 5.2.6

To write $\frac{512}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{10} (\frac{512}{5} \cdot \frac{2}{2} - \frac{1}{10})$

Step 5.2.7

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.7.1

Multiply $\frac{512}{5}$ by $\frac{2}{2}$ .

$\frac{1}{10} (\frac{512 \cdot 2}{5 \cdot 2} - \frac{1}{10})$

Step 5.2.7.2

Multiply $5$ by $2$ .

$\frac{1}{10} (\frac{512 \cdot 2}{10} - \frac{1}{10})$

Step 5.2.8

Combine the numerators over the common denominator.

$\frac{1}{10} \cdot \frac{512 \cdot 2 - 1}{10}$

Step 5.2.9

Simplify the numerator.

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Step 5.2.9.1

Multiply $512$ by $2$ .

$\frac{1}{10} \cdot \frac{1024 - 1}{10}$

Step 5.2.9.2

Subtract $1$ from $1024$ .

$\frac{1}{10} \cdot \frac{1023}{10}$

Step 5.2.10

Multiply $\frac{1}{10}$ by $\frac{1023}{10}$ .

$\frac{1023}{10 \cdot 10}$

Step 5.2.11

Multiply $10$ by $10$ .

$\frac{1023}{100}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1023}{100}$

Decimal Form:

$10.23$

Mixed Number Form:

$10 \frac{23}{100}$

Step 7