Calculus Examples

$\frac{d x}{d y} = \frac{y^{2}}{{(1 - x^{2})}^{\frac{1}{2}}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by ${(1 - x^{2})}^{\frac{1}{2}}$ .

${(1 - x^{2})}^{\frac{1}{2}} \frac{d x}{d y} = {(1 - x^{2})}^{\frac{1}{2}} \frac{y^{2}}{{(1 - x^{2})}^{\frac{1}{2}}}$

Step 1.2

Cancel the common factor of ${(1 - x^{2})}^{\frac{1}{2}}$ .

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Step 1.2.1

Cancel the common factor.

${(1 - x^{2})}^{\frac{1}{2}} \frac{d x}{d y} = {(1 - x^{2})}^{\frac{1}{2}} \frac{y^{2}}{{(1 - x^{2})}^{\frac{1}{2}}}$

Step 1.2.2

Rewrite the expression.

${(1 - x^{2})}^{\frac{1}{2}} \frac{d x}{d y} = y^{2}$

Step 1.3

Rewrite the equation.

${(1 - x^{2})}^{\frac{1}{2}} d x = y^{2} d y$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int {(1 - x^{2})}^{\frac{1}{2}} d x = \int y^{2} d y$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 2.2.2

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.

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

Step 2.2.3

Simplify terms.

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Step 2.2.3.1

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

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Step 2.2.3.1.1

Apply pythagorean identity.

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

Step 2.2.3.1.2

Multiply the exponents in ${(\cos^{2} (t))}^{1}$ .

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Step 2.2.3.1.2.1

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

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

Step 2.2.3.1.2.2

Multiply $2$ by $1$ .

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

Step 2.2.3.1.3

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

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

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

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

Step 2.2.3.2.3

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

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

Step 2.2.3.2.4

Add $1$ and $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Split the single integral into multiple integrals.

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

Step 2.2.7

Apply the constant rule.

$\frac{1}{2} (t + C_{1} + \int \cos (2 t) d t) = \int y^{2} d y$

Step 2.2.8

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 2.2.8.1

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

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Step 2.2.8.1.1

Differentiate $2 t$ .

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

Step 2.2.8.1.2

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 2.2.8.1.3

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 2.2.8.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.8.2

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

$\frac{1}{2} (t + C_{1} + \int \cos (u) \frac{1}{2} d u) = \int y^{2} d y$

Step 2.2.9

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

$\frac{1}{2} (t + C_{1} + \int \frac{\cos (u)}{2} d u) = \int y^{2} d y$

Step 2.2.10

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

$\frac{1}{2} (t + C_{1} + \frac{1}{2} \int \cos (u) d u) = \int y^{2} d y$

Step 2.2.11

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

$\frac{1}{2} (t + C_{1} + \frac{1}{2} (\sin (u) + C_{2})) = \int y^{2} d y$

Step 2.2.12

Simplify.

$\frac{1}{2} (t + \frac{1}{2} \sin (u)) + C_{3} = \int y^{2} d y$

Step 2.2.13

Substitute back in for each integration substitution variable.

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Step 2.2.13.1

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

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (u)) + C_{3} = \int y^{2} d y$

Step 2.2.13.2

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

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (2 t)) + C_{3} = \int y^{2} d y$

Step 2.2.13.3

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

$\frac{1}{2} (\arcsin (x) + \frac{1}{2} \sin (2 \arcsin (x))) + C_{3} = \int y^{2} d y$

Step 2.2.14

Simplify.

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Step 2.2.14.1

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

$\frac{1}{2} (\arcsin (x) + \frac{\sin (2 \arcsin (x))}{2}) + C_{3} = \int y^{2} d y$

Step 2.2.14.2

Apply the distributive property.

$\frac{1}{2} \arcsin (x) + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x))}{2} + C_{3} = \int y^{2} d y$

Step 2.2.14.3

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

$\frac{\arcsin (x)}{2} + \frac{1}{2} \cdot \frac{\sin (2 \arcsin (x))}{2} + C_{3} = \int y^{2} d y$

Step 2.2.14.4

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

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Step 2.2.14.4.1

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

$\frac{\arcsin (x)}{2} + \frac{\sin (2 \arcsin (x))}{2 \cdot 2} + C_{3} = \int y^{2} d y$

Step 2.2.14.4.2

Multiply $2$ by $2$ .

$\frac{\arcsin (x)}{2} + \frac{\sin (2 \arcsin (x))}{4} + C_{3} = \int y^{2} d y$

Step 2.2.15

Reorder terms.

$\frac{1}{2} \arcsin (x) + \frac{1}{4} \sin (2 \arcsin (x)) + C_{3} = \int y^{2} d y$

Step 2.3

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$\frac{1}{2} \arcsin (x) + \frac{1}{4} \sin (2 \arcsin (x)) + C_{3} = \frac{1}{3} y^{3} + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{2} \arcsin (x) + \frac{1}{4} \sin (2 \arcsin (x)) = \frac{1}{3} y^{3} + K$