Calculus Examples

$\frac{\arcsin (x)}{y} d x + (1 - e^{y}) d y = 0$

Step 1

Subtract $\frac{\arcsin (x)}{y} d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $y$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(y \cdot 1 + y (- e^{y})) d y = y (- \frac{\arcsin (x)}{y}) d x$

Step 3.2

Multiply $y$ by $1$ .

$(y + y (- e^{y})) d y = y (- \frac{\arcsin (x)}{y}) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$(y - y e^{y}) d y = y (- \frac{\arcsin (x)}{y}) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$(y - y e^{y}) d y = - y \frac{\arcsin (x)}{y} d x$

Step 3.5

Cancel the common factor of $y$ .

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Step 3.5.1

Factor $y$ out of $- y$ .

$(y - y e^{y}) d y = y \cdot - 1 \frac{\arcsin (x)}{y} d x$

Step 3.5.2

Cancel the common factor.

$(y - y e^{y}) d y = y \cdot - 1 \frac{\arcsin (x)}{y} d x$

Step 3.5.3

Rewrite the expression.

$(y - y e^{y}) d y = - \arcsin (x) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int y - y e^{y} d y = \int - \arcsin (x) d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the single integral into multiple integrals.

$\int y d y + \int - y e^{y} d y = \int - \arcsin (x) d x$

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{y}$ .

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

Step 4.2.5

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

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

Step 4.2.6

Simplify.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arcsin (x)$ and $d v = 1$ .

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

Step 4.3.3

Combine $x$ and $\frac{1}{\sqrt{1 - x^{2}}}$ .

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

Step 4.3.4

Let $u = 1 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.4.1

Let $u = 1 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 4.3.4.1.1

Differentiate $1 - x^{2}$ .

$\frac{d}{d x} [1 - x^{2}]$

Step 4.3.4.1.2

Differentiate.

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Step 4.3.4.1.2.1

By the Sum Rule, the derivative of $1 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$

Step 4.3.4.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 4.3.4.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 4.3.4.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 4.3.4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 4.3.4.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 4.3.4.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 4.3.4.2

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

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

Step 4.3.5

Simplify.

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Step 4.3.5.1

Move the negative in front of the fraction.

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

Step 4.3.5.2

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

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

Step 4.3.5.3

Move $2$ to the left of $\sqrt{u}$ .

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

Step 4.3.6

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

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

Step 4.3.7

Simplify.

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Step 4.3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.7.2

Multiply $\int \frac{1}{2 \sqrt{u}} d u$ by $1$ .

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

Step 4.3.8

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

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

Step 4.3.9

Apply basic rules of exponents.

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Step 4.3.9.1

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

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

Step 4.3.9.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3.9.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 4.3.9.3.1

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

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

Step 4.3.9.3.2

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

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

Step 4.3.9.3.3

Move the negative in front of the fraction.

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

Step 4.3.10

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

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

Step 4.3.11

Rewrite $- (\arcsin (x) x + \frac{1}{2} (2 u^{\frac{1}{2}} + C_{4}))$ as $- (\arcsin (x) x + u^{\frac{1}{2}}) + C_{4}$ .

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

Step 4.3.12

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 4.3.13

Simplify.

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Step 4.3.13.1

Apply the distributive property.

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

Step 4.3.13.2

Reorder factors in $- \arcsin (x) x - {(1 - x^{2})}^{\frac{1}{2}}$ .

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

Step 4.4

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

$\frac{1}{2} y^{2} - (y e^{y} - e^{y}) = - x \arcsin (x) - {(1 - x^{2})}^{\frac{1}{2}} + K$