Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 - x$ .

$\frac{d}{d x} [1 - x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [- x]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

Step 1.1.3.2

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

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 0$

Step 1.3

Subtract $0$ from $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

$u_{upper} = 1 - 1 \cdot 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 1.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{1}^{0} - u^{9} d u$

Step 2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{1}^{0} u^{9} d u$

Step 3

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

$- (\frac{1}{10} u^{10}]_{1}^{0})$

Step 4

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

$- (\frac{u^{10}}{10}]_{1}^{0})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{u^{10}}{10}$ at $0$ and at $1$ .

$- ((\frac{0^{10}}{10}) - \frac{1^{10}}{10})$

Step 5.2

Simplify.

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Step 5.2.1

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

$- (\frac{0}{10} - \frac{1^{10}}{10})$

Step 5.2.2

Cancel the common factor of $0$ and $10$ .

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Step 5.2.2.1

Factor $10$ out of $0$ .

$- (\frac{10 (0)}{10} - \frac{1^{10}}{10})$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $10$ out of $10$ .

$- (\frac{10 \cdot 0}{10 \cdot 1} - \frac{1^{10}}{10})$

Step 5.2.2.2.2

Cancel the common factor.

$- (\frac{10 \cdot 0}{10 \cdot 1} - \frac{1^{10}}{10})$

Step 5.2.2.2.3

Rewrite the expression.

$- (\frac{0}{1} - \frac{1^{10}}{10})$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$- (0 - \frac{1^{10}}{10})$

Step 5.2.3

One to any power is one.

$- (0 - \frac{1}{10})$

Step 5.2.4

Subtract $\frac{1}{10}$ from $0$ .

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

Step 5.2.5

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{10})$

Step 5.2.6

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

$\frac{1}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{10}$

Decimal Form:

$0.1$

Step 7