Calculus Examples

$\sqrt{1 - x^{2}}$

Step 1

Write $\sqrt{1 - x^{2}}$ as a function.

$f (x) = \sqrt{1 - x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sqrt{1 - x^{2}} d x$

Step 4

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.

$\int \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Apply pythagorean identity.

$\int \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 5.1.2

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

$\int \cos (t) \cos (t) d t$

Step 5.2

Simplify.

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Step 5.2.1

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

$\int \cos^{1} (t) \cos (t) d t$

Step 5.2.2

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

$\int \cos^{1} (t) \cos^{1} (t) d t$

Step 5.2.3

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

$\int {\cos (t)}^{1 + 1} d t$

Step 5.2.4

Add $1$ and $1$ .

$\int \cos^{2} (t) d t$

Step 6

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

$\int \frac{1 + \cos (2 t)}{2} d t$

Step 7

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

$\frac{1}{2} \int 1 + \cos (2 t) d t$

Step 8

Split the single integral into multiple integrals.

$\frac{1}{2} (\int d t + \int \cos (2 t) d t)$

Step 9

Apply the constant rule.

$\frac{1}{2} (t + C + \int \cos (2 t) d t)$

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

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 11

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

$\frac{1}{2} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 12

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

$\frac{1}{2} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 13

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

$\frac{1}{2} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 14

Simplify.

$\frac{1}{2} (t + \frac{1}{2} \sin (u)) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $t$ with $\arcsin (x)$ .

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (u)) + C$

Step 15.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (2 t)) + C$

Step 15.3

Replace all occurrences of $t$ with $\arcsin (x)$ .

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (2 \arcsin (x))) + C$

Step 16

Simplify.

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Step 16.1

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

$\frac{1}{2} (\arcsin (x) + \frac{\sin (2 \arcsin (x))}{2}) + C$

Step 16.2

Apply the distributive property.

$\frac{1}{2} \arcsin (x) + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x))}{2} + C$

Step 16.3

Combine $\frac{1}{2}$ and $\arcsin (x)$ .

$\frac{\arcsin (x)}{2} + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x))}{2} + C$

Step 16.4

Multiply $\frac{1}{2} \cdot \frac{\sin (2 \arcsin (x))}{2}$ .

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Step 16.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 \arcsin (x))}{2}$ .

$\frac{\arcsin (x)}{2} + \frac{\sin (2 \arcsin (x))}{2 \cdot 2} + C$

Step 16.4.2

Multiply $2$ by $2$ .

$\frac{\arcsin (x)}{2} + \frac{\sin (2 \arcsin (x))}{4} + C$

Step 17

Reorder terms.

$\frac{1}{2} \arcsin (x) + \frac{1}{4} \sin (2 \arcsin (x)) + C$

Step 18

The answer is the antiderivative of the function $f (x) = \sqrt{1 - x^{2}}$ .

$F (x) =$ $\frac{1}{2} \arcsin (x) + \frac{1}{4} \sin (2 \arcsin (x)) + C$