Calculus Examples

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

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.

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

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.

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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 8.1

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

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Step 8.1.1

Differentiate $2 t$ .

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

$u_{lower} = 2 \cdot 0$

Step 8.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 8.4

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

$u_{upper} = 2 \frac{π}{4}$

Step 8.5

Cancel the common factor of $2$ .

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Step 8.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 8.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 8.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 8.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 8.7

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

$\frac{1}{2} (t]_{0}^{\frac{π}{4}} - \frac{1}{2} \sin (u)]_{0}^{\frac{π}{2}})$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $t$ at $\frac{π}{4}$ and at $0$ .

$\frac{1}{2} ((\frac{π}{4}) + 0 - \frac{1}{2} \sin (u)]_{0}^{\frac{π}{2}})$

Step 12.2

Evaluate $\sin (u)$ at $\frac{π}{2}$ and at $0$ .

$\frac{1}{2} ((\frac{π}{4}) + 0 - \frac{1}{2} ((\sin (\frac{π}{2})) - \sin (0)))$

Step 12.3

Add $\frac{π}{4}$ and $0$ .

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2} (\sin (\frac{π}{2}) - \sin (0)))$

Step 13

Simplify.

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Step 13.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2} (1 - \sin (0)))$

Step 13.2

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

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2} (1 - 0))$

Step 13.3

Multiply $- 1$ by $0$ .

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2} (1 + 0))$

Step 13.4

Add $1$ and $0$ .

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2} \cdot 1)$

Step 13.5

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\frac{π}{4} - \frac{1}{2})$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{1}{2} \cdot \frac{π}{4} + \frac{1}{2} (- \frac{1}{2})$

Step 14.2

Multiply $\frac{1}{2} \cdot \frac{π}{4}$ .

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Step 14.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$\frac{π}{2 \cdot 4} + \frac{1}{2} (- \frac{1}{2})$

Step 14.2.2

Multiply $2$ by $4$ .

$\frac{π}{8} + \frac{1}{2} (- \frac{1}{2})$

Step 14.3

Multiply $\frac{1}{2} (- \frac{1}{2})$ .

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Step 14.3.1

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

$\frac{π}{8} - \frac{1}{2 \cdot 2}$

Step 14.3.2

Multiply $2$ by $2$ .

$\frac{π}{8} - \frac{1}{4}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{8} - \frac{1}{4}$

Decimal Form:

$0.14269908 \dots$

Step 16