Calculus Examples

$8 \int_{0}^{1} x^{3} \sqrt{1 - x^{2}} d x$

Step 1

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{1 - \sin^{2} (t)}$ .

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Step 2.1.1

Apply pythagorean identity.

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) \cos (t) \cos (t) d t$

Step 2.2

Simplify.

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Step 2.2.1

Raise $\cos (t)$ to the power of $1$ .

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) (\cos^{1} (t) \cos (t)) d t$

Step 2.2.2

Raise $\cos (t)$ to the power of $1$ .

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) (\cos^{1} (t) \cos^{1} (t)) d t$

Step 2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) {\cos (t)}^{1 + 1} d t$

Step 2.2.4

Add $1$ and $1$ .

$8 \int_{0}^{\frac{π}{2}} \sin^{3} (t) \cos^{2} (t) d t$

Step 3

Factor out $\sin^{2} (t)$ .

$8 \int_{0}^{\frac{π}{2}} \sin^{2} (t) \sin (t) \cos^{2} (t) d t$

Step 4

Using the Pythagorean Identity, rewrite $\sin^{2} (t)$ as $1 - \cos^{2} (t)$ .

$8 \int_{0}^{\frac{π}{2}} (1 - \cos^{2} (t)) \sin (t) \cos^{2} (t) d t$

Step 5

Let $u = \cos (t)$ . Then $d u = - \sin (t) d t$ , so $- \frac{1}{\sin (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = \cos (t)$ . Find $\frac{d u}{d t}$ .

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Step 5.1.1

Differentiate $\cos (t)$ .

$\frac{d}{d t} [\cos (t)]$

Step 5.1.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$- \sin (t)$

Step 5.2

Substitute the lower limit in for $t$ in $u = \cos (t)$ .

$u_{lower} = \cos (0)$

Step 5.3

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for $t$ in $u = \cos (t)$ .

$u_{upper} = \cos (\frac{π}{2})$

Step 5.5

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{upper} = 0$

Step 5.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 5.7

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

$8 \int_{1}^{0} (- 1 + u^{2}) u^{2} d u$

Step 6

Multiply $(- 1 + u^{2}) u^{2}$ .

$8 \int_{1}^{0} - 1 u^{2} + u^{2} u^{2} d u$

Step 7

Simplify.

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Step 7.1

Rewrite $- 1 u^{2}$ as $- u^{2}$ .

$8 \int_{1}^{0} - u^{2} + u^{2} u^{2} d u$

Step 7.2

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 7.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 \int_{1}^{0} - u^{2} + u^{2 + 2} d u$

Step 7.2.2

Add $2$ and $2$ .

$8 \int_{1}^{0} - u^{2} + u^{4} d u$

Step 8

Split the single integral into multiple integrals.

$8 (\int_{1}^{0} - u^{2} d u + \int_{1}^{0} u^{4} d u)$

Step 9

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

$8 (- \int_{1}^{0} u^{2} d u + \int_{1}^{0} u^{4} d u)$

Step 10

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$8 (- (\frac{1}{3} u^{3}]_{1}^{0}) + \int_{1}^{0} u^{4} d u)$

Step 11

Combine $\frac{1}{3}$ and $u^{3}$ .

$8 (- (\frac{u^{3}}{3}]_{1}^{0}) + \int_{1}^{0} u^{4} d u)$

Step 12

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$8 (- (\frac{u^{3}}{3}]_{1}^{0}) + \frac{1}{5} u^{5}]_{1}^{0})$

Step 13

Combine $\frac{1}{5}$ and $u^{5}$ .

$8 (- (\frac{u^{3}}{3}]_{1}^{0}) + \frac{u^{5}}{5}]_{1}^{0})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\frac{u^{3}}{3}$ at $0$ and at $1$ .

$8 (- ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}) + \frac{u^{5}}{5}]_{1}^{0})$

Step 14.2

Evaluate $\frac{u^{5}}{5}$ at $0$ and at $1$ .

$8 (- (\frac{0^{3}}{3} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3

Simplify.

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Step 14.3.1

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

$8 (- (\frac{0}{3} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.2

Cancel the common factor of $0$ and $3$ .

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Step 14.3.2.1

Factor $3$ out of $0$ .

$8 (- (\frac{3 (0)}{3} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.2.2

Cancel the common factors.

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Step 14.3.2.2.1

Factor $3$ out of $3$ .

$8 (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.2.2.2

Cancel the common factor.

$8 (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.2.2.3

Rewrite the expression.

$8 (- (\frac{0}{1} - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.2.2.4

Divide $0$ by $1$ .

$8 (- (0 - \frac{1^{3}}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.3

One to any power is one.

$8 (- (0 - \frac{1}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.4

Subtract $\frac{1}{3}$ from $0$ .

$8 (- - \frac{1}{3} + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.5

Multiply $- 1$ by $- 1$ .

$8 (1 (\frac{1}{3}) + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.6

Multiply $\frac{1}{3}$ by $1$ .

$8 (\frac{1}{3} + \frac{0^{5}}{5} - \frac{1^{5}}{5})$

Step 14.3.7

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

$8 (\frac{1}{3} + \frac{0}{5} - \frac{1^{5}}{5})$

Step 14.3.8

Cancel the common factor of $0$ and $5$ .

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Step 14.3.8.1

Factor $5$ out of $0$ .

$8 (\frac{1}{3} + \frac{5 (0)}{5} - \frac{1^{5}}{5})$

Step 14.3.8.2

Cancel the common factors.

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Step 14.3.8.2.1

Factor $5$ out of $5$ .

$8 (\frac{1}{3} + \frac{5 \cdot 0}{5 \cdot 1} - \frac{1^{5}}{5})$

Step 14.3.8.2.2

Cancel the common factor.

$8 (\frac{1}{3} + \frac{5 \cdot 0}{5 \cdot 1} - \frac{1^{5}}{5})$

Step 14.3.8.2.3

Rewrite the expression.

$8 (\frac{1}{3} + \frac{0}{1} - \frac{1^{5}}{5})$

Step 14.3.8.2.4

Divide $0$ by $1$ .

$8 (\frac{1}{3} + 0 - \frac{1^{5}}{5})$

Step 14.3.9

One to any power is one.

$8 (\frac{1}{3} + 0 - \frac{1}{5})$

Step 14.3.10

Subtract $\frac{1}{5}$ from $0$ .

$8 (\frac{1}{3} - \frac{1}{5})$

Step 14.3.11

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$8 (\frac{1}{3} \cdot \frac{5}{5} - \frac{1}{5})$

Step 14.3.12

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 (\frac{1}{3} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{3}{3})$

Step 14.3.13

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.13.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$8 (\frac{5}{3 \cdot 5} - \frac{1}{5} \cdot \frac{3}{3})$

Step 14.3.13.2

Multiply $3$ by $5$ .

$8 (\frac{5}{15} - \frac{1}{5} \cdot \frac{3}{3})$

Step 14.3.13.3

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$8 (\frac{5}{15} - \frac{3}{5 \cdot 3})$

Step 14.3.13.4

Multiply $5$ by $3$ .

$8 (\frac{5}{15} - \frac{3}{15})$

Step 14.3.14

Combine the numerators over the common denominator.

$8 \frac{5 - 3}{15}$

Step 14.3.15

Subtract $3$ from $5$ .

$8 (\frac{2}{15})$

Step 14.3.16

Combine $8$ and $\frac{2}{15}$ .

$\frac{8 \cdot 2}{15}$

Step 14.3.17

Multiply $8$ by $2$ .

$\frac{16}{15}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{15}$

Decimal Form:

$1.0\overline{6}$

Mixed Number Form:

$1 \frac{1}{15}$

Step 16