Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Multiply by $1$ .

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Apply pythagorean identity.

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

Step 5.1.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\int \sqrt{{(2^{\frac{1}{2}})}^{2} \cdot \cos^{2} (t)} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{2^{\frac{1}{2} \cdot 2} \cdot \cos^{2} (t)} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.3

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

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

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

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

Step 5.1.6.4.2

Rewrite the expression.

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

Step 5.1.6.5

Evaluate the exponent.

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

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

Step 5.1.7

Reorder $2$ and $\cos^{2} (t)$ .

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

Step 5.1.8

Pull terms out from under the radical.

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

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

Step 5.2.5

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.6

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.7

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

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

Step 5.2.8

Add $1$ and $1$ .

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

Step 5.2.9

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.2.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 5.2.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2.9.3

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

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

Step 5.2.9.4

Cancel the common factor of $2$ .

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Step 5.2.9.4.1

Cancel the common factor.

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

Step 5.2.9.4.2

Rewrite the expression.

$\int \cos^{2} (t) \cdot 2^{1} d t$

Step 5.2.9.5

Evaluate the exponent.

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

Step 5.2.10

Move $2$ to the left of $\cos^{2} (t)$ .

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

Step 6

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

$1 \int 1 + \cos (2 t) d t$

Step 9.3

Multiply $\int 1 + \cos (2 t) d t$ by $1$ .

$\int 1 + \cos (2 t) d t$

Step 10

Split the single integral into multiple integrals.

$\int d t + \int \cos (2 t) d t$

Step 11

Apply the constant rule.

$t + C + \int \cos (2 t) d t$

Step 12

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 12.1

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

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Step 12.1.1

Differentiate $2 t$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

Replace all occurrences of $t$ with $\arcsin (\frac{x}{\sqrt{2}})$ .

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

Step 17.2

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

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

Step 17.3

Replace all occurrences of $t$ with $\arcsin (\frac{x}{\sqrt{2}})$ .

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

Step 18

Simplify.

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Step 18.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 18.2

Combine and simplify the denominator.

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Step 18.2.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 18.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 18.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 18.2.4

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

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

Step 18.2.5

Add $1$ and $1$ .

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

Step 18.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 18.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 18.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 18.2.6.3

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

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

Step 18.2.6.4

Cancel the common factor of $2$ .

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Step 18.2.6.4.1

Cancel the common factor.

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

Step 18.2.6.4.2

Rewrite the expression.

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

Step 18.2.6.5

Evaluate the exponent.

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

Step 18.3

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 18.4

Combine and simplify the denominator.

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Step 18.4.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 18.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 18.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 18.4.4

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

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

Step 18.4.5

Add $1$ and $1$ .

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

Step 18.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 18.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 18.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 18.4.6.3

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

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

Step 18.4.6.4

Cancel the common factor of $2$ .

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Step 18.4.6.4.1

Cancel the common factor.

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

Step 18.4.6.4.2

Rewrite the expression.

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

Step 18.4.6.5

Evaluate the exponent.

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

Step 18.5

Reorder terms.

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

Step 19

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

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