Calculus Examples

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

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.

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

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.

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

Step 2.1.2

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Add $1$ and $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $2 t$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

$u_{lower} = 2 \cdot 0$

Step 7.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 7.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 7.5.2

Rewrite the expression.

$u_{upper} = π$

Step 7.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 7.7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $t$ at $\frac{π}{2}$ and at $0$ .

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

Step 11.2

Evaluate $\sin (u)$ at $π$ and at $0$ .

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

Step 11.3

Add $\frac{π}{2}$ and $0$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Multiply $- 1$ by $0$ .

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

Step 12.3

Add $\sin (π)$ and $0$ .

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

Step 12.4

Combine $\frac{1}{2}$ and $\sin (π)$ .

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

Step 13

Simplify.

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Step 13.1

Combine the numerators over the common denominator.

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

Step 13.2

Simplify each term.

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Step 13.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 13.2.2

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

$\frac{1}{2} \cdot \frac{π + 0}{2}$

Step 13.3

Add $π$ and $0$ .

$\frac{1}{2} \cdot \frac{π}{2}$

Step 13.4

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

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Step 13.4.1

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

$\frac{π}{2 \cdot 2}$

Step 13.4.2

Multiply $2$ by $2$ .

$\frac{π}{4}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{4}$

Decimal Form:

$0.78539816 \dots$

Step 15