Calculus Examples

$\int_{0}^{1} 9 x {(x^{2} - 1)}^{9} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{0}^{1} x {(x^{2} - 1)}^{9} d x$

Step 2

Let $u = x^{2} - 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x^{2} - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} - 1$ .

$\frac{d}{d x} [x^{2} - 1]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 1]$

Step 2.1.4

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

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

Substitute the lower limit in for $x$ in $u = x^{2} - 1$ .

$u_{lower} = 0^{2} - 1$

Step 2.3

Simplify.

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Step 2.3.1

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

$u_{lower} = 0 - 1$

Step 2.3.2

Subtract $1$ from $0$ .

$u_{lower} = - 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = x^{2} - 1$ .

$u_{upper} = 1^{2} - 1$

Step 2.5

Simplify.

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Step 2.5.1

One to any power is one.

$u_{upper} = 1 - 1$

Step 2.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 2.6

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

$u_{lower} = - 1$

$u_{upper} = 0$

Step 2.7

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

$9 \int_{- 1}^{0} u^{9} \frac{1}{2} d u$

Step 3

Combine $u^{9}$ and $\frac{1}{2}$ .

$9 \int_{- 1}^{0} \frac{u^{9}}{2} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$9 (\frac{1}{2} \int_{- 1}^{0} u^{9} d u)$

Step 5

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int_{- 1}^{0} u^{9} d u$

Step 6

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

$\frac{9}{2} \frac{1}{10} u^{10}]_{- 1}^{0}$

Step 7

Substitute and simplify.

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Step 7.1

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

$\frac{9}{2} ((\frac{1}{10} \cdot 0^{10}) - \frac{1}{10} {(- 1)}^{10})$

Step 7.2

Simplify.

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Step 7.2.1

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

$\frac{9}{2} (\frac{1}{10} \cdot 0 - \frac{1}{10} {(- 1)}^{10})$

Step 7.2.2

Multiply $\frac{1}{10}$ by $0$ .

$\frac{9}{2} (0 - \frac{1}{10} {(- 1)}^{10})$

Step 7.2.3

Raise $- 1$ to the power of $10$ .

$\frac{9}{2} (0 - \frac{1}{10} \cdot 1)$

Step 7.2.4

Multiply $- 1$ by $1$ .

$\frac{9}{2} (0 - \frac{1}{10})$

Step 7.2.5

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

$\frac{9}{2} (- \frac{1}{10})$

Step 7.2.6

Multiply $\frac{9}{2}$ by $\frac{1}{10}$ .

$- \frac{9}{2 \cdot 10}$

Step 7.2.7

Multiply $2$ by $10$ .

$- \frac{9}{20}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{9}{20}$

Decimal Form:

$- 0.45$

Step 9