Calculus Examples

$\int_{0}^{1} \frac{x^{2}}{\sqrt{4 - x^{2}}} d x$

Step 1

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

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 - {(2 \sin (t))}^{2}}} (2 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 - {(2 \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $2 \sin (t)$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 - (2^{2} \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.1.2

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

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 - (4 \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.1.3

Multiply $4$ by $- 1$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 - 4 \sin^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.2

Factor $4$ out of $4$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 (1) - 4 \sin^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 (1) + 4 (- \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 (1 - \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{4 \cos^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.6

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{\sqrt{{(2 \cos (t))}^{2}}} (2 \cos (t)) d t$

Step 2.1.7

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

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{2 \cos (t)} (2 \cos (t)) d t$

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

Cancel the common factor of $2 \cos (t)$ .

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Step 2.2.1.1

Cancel the common factor.

$\int_{0}^{\frac{π}{6}} \frac{{(2 \sin (t))}^{2}}{2 \cos (t)} (2 \cos (t)) d t$

Step 2.2.1.2

Rewrite the expression.

$\int_{0}^{\frac{π}{6}} {(2 \sin (t))}^{2} d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Factor $2$ out of $2 \sin (t)$ .

$\int_{0}^{\frac{π}{6}} {(2 (\sin (t)))}^{2} d t$

Step 2.2.2.2

Apply the product rule to $2 (\sin (t))$ .

$\int_{0}^{\frac{π}{6}} 2^{2} \sin^{2} (t) d t$

Step 2.2.2.3

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

$\int_{0}^{\frac{π}{6}} 4 \sin^{2} (t) d t$

Step 3

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int_{0}^{\frac{π}{6}} \sin^{2} (t) d t$

Step 4

Use the half-angle formula to rewrite $\sin^{2} (t)$ as $\frac{1 - \cos (2 t)}{2}$ .

$4 \int_{0}^{\frac{π}{6}} \frac{1 - \cos (2 t)}{2} d t$

Step 5

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

$4 (\frac{1}{2} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 6.2

Cancel the common factor of $4$ and $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 6.2.2.3

Rewrite the expression.

$\frac{2}{1} \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 6.2.2.4

Divide $2$ by $1$ .

$2 \int_{0}^{\frac{π}{6}} 1 - \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$2 (\int_{0}^{\frac{π}{6}} d t + \int_{0}^{\frac{π}{6}} - \cos (2 t) d t)$

Step 8

Apply the constant rule.

$2 (t]_{0}^{\frac{π}{6}} + \int_{0}^{\frac{π}{6}} - \cos (2 t) d t)$

Step 9

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

$2 (t]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} \cos (2 t) d t)$

Step 10

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

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Step 10.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 10.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 10.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 10.1.3

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

$2 \cdot 1$

Step 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 \cdot 0$

Step 10.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 10.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \frac{π}{6}$

Step 10.5

Cancel the common factor of $2$ .

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Step 10.5.1

Factor $2$ out of $6$ .

$u_{upper} = 2 \frac{π}{2 (3)}$

Step 10.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 3}$

Step 10.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 10.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{3}$

Step 10.7

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

$2 (t]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{3}} \cos (u) \frac{1}{2} d u)$

Step 11

Combine $\cos (u)$ and $\frac{1}{2}$ .

$2 (t]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{3}} \frac{\cos (u)}{2} d u)$

Step 12

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

$2 (t]_{0}^{\frac{π}{6}} - (\frac{1}{2} \int_{0}^{\frac{π}{3}} \cos (u) d u))$

Step 13

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$2 (t]_{0}^{\frac{π}{6}} - \frac{1}{2} \sin (u)]_{0}^{\frac{π}{3}})$

Step 14

Simplify.

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Step 14.1

Evaluate $t$ at $\frac{π}{6}$ and at $0$ .

$2 ((\frac{π}{6}) + 0 - \frac{1}{2} \sin (u)]_{0}^{\frac{π}{3}})$

Step 14.2

Simplify the expression.

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Step 14.2.1

Evaluate $\sin (u)$ at $\frac{π}{3}$ and at $0$ .

$2 ((\frac{π}{6}) + 0 - \frac{1}{2} ((\sin (\frac{π}{3})) - \sin (0)))$

Step 14.2.2

Add $\frac{π}{6}$ and $0$ .

$2 (\frac{π}{6} - \frac{1}{2} ((\sin (\frac{π}{3})) - \sin (0)))$

Step 14.3

Simplify.

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Step 14.3.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$2 (\frac{π}{6} - \frac{1}{2} (\frac{\sqrt{3}}{2} - \sin (0)))$

Step 14.3.2

The exact value of $\sin (0)$ is $0$ .

$2 (\frac{π}{6} - \frac{1}{2} (\frac{\sqrt{3}}{2} - 0))$

Step 14.3.3

Multiply $- 1$ by $0$ .

$2 (\frac{π}{6} - \frac{1}{2} (\frac{\sqrt{3}}{2} + 0))$

Step 14.3.4

Add $\frac{\sqrt{3}}{2}$ and $0$ .

$2 (\frac{π}{6} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2})$

Step 14.3.5

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{2}$ .

$2 (\frac{π}{6} - \frac{\sqrt{3}}{2 \cdot 2})$

Step 14.3.6

Multiply $2$ by $2$ .

$2 (\frac{π}{6} - \frac{\sqrt{3}}{4})$

Step 14.4

Simplify.

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Step 14.4.1

Apply the distributive property.

$2 \frac{π}{6} + 2 (- \frac{\sqrt{3}}{4})$

Step 14.4.2

Cancel the common factor of $2$ .

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Step 14.4.2.1

Factor $2$ out of $6$ .

$2 \frac{π}{2 (3)} + 2 (- \frac{\sqrt{3}}{4})$

Step 14.4.2.2

Cancel the common factor.

$2 \frac{π}{2 \cdot 3} + 2 (- \frac{\sqrt{3}}{4})$

Step 14.4.2.3

Rewrite the expression.

$\frac{π}{3} + 2 (- \frac{\sqrt{3}}{4})$

Step 14.4.3

Cancel the common factor of $2$ .

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Step 14.4.3.1

Move the leading negative in $- \frac{\sqrt{3}}{4}$ into the numerator.

$\frac{π}{3} + 2 \frac{- \sqrt{3}}{4}$

Step 14.4.3.2

Factor $2$ out of $4$ .

$\frac{π}{3} + 2 \frac{- \sqrt{3}}{2 (2)}$

Step 14.4.3.3

Cancel the common factor.

$\frac{π}{3} + 2 \frac{- \sqrt{3}}{2 \cdot 2}$

Step 14.4.3.4

Rewrite the expression.

$\frac{π}{3} + \frac{- \sqrt{3}}{2}$

Step 14.4.4

Move the negative in front of the fraction.

$\frac{π}{3} - \frac{\sqrt{3}}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{3} - \frac{\sqrt{3}}{2}$

Decimal Form:

$0.18117214 \dots$