Calculus Examples

$x y^{4} d x + (y^{2} + 2) e^{x} d y = 0$

Step 1

Subtract $x y^{4} d x$ from both sides of the equation.

$(y^{2} + 2) e^{x} d y = - x y^{4} d x$

Step 2

Multiply both sides by $\frac{1}{e^{x} y^{4}}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $e^{x}$ .

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Step 3.1.1

Factor $e^{x}$ out of $e^{x} y^{4}$ .

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

Step 3.1.2

Factor $e^{x}$ out of $(y^{2} + 2) e^{x}$ .

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

Step 3.1.3

Cancel the common factor.

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

Step 3.1.4

Rewrite the expression.

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

Step 3.2

Multiply $\frac{1}{y^{4}}$ by $y^{2} + 2$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $y^{4}$ .

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Step 3.4.1

Move the leading negative in $- \frac{1}{e^{x} y^{4}}$ into the numerator.

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

Step 3.4.2

Factor $y^{4}$ out of $e^{x} y^{4}$ .

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

Step 3.4.3

Factor $y^{4}$ out of $x y^{4}$ .

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

$\frac{y^{2} + 2}{y^{4}} d y = \frac{- 1}{e^{x}} x d x$

Step 3.5

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

$\frac{y^{2} + 2}{y^{4}} d y = \frac{- x}{e^{x}} d x$

Step 3.6

Move the negative in front of the fraction.

$\frac{y^{2} + 2}{y^{4}} d y = - \frac{x}{e^{x}} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{y^{2} + 2}{y^{4}} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Apply basic rules of exponents.

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Step 4.2.1.1

Move $y^{4}$ out of the denominator by raising it to the $- 1$ power.

$\int (y^{2} + 2) {(y^{4})}^{- 1} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.1.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 4.2.1.2.1

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

$\int (y^{2} + 2) y^{4 \cdot - 1} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.1.2.2

Multiply $4$ by $- 1$ .

$\int (y^{2} + 2) y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.2

Multiply $(y^{2} + 2) y^{- 4}$ .

$\int y^{2} y^{- 4} + 2 y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.3

Multiply $y^{2}$ by $y^{- 4}$ by adding the exponents.

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Step 4.2.3.1

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

$\int y^{2 - 4} + 2 y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.3.2

Subtract $4$ from $2$ .

$\int y^{- 2} + 2 y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.4

Split the single integral into multiple integrals.

$\int y^{- 2} d y + \int 2 y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.5

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

$- y^{- 1} + C_{1} + \int 2 y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.6

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

$- y^{- 1} + C_{1} + 2 \int y^{- 4} d y = \int - \frac{x}{e^{x}} d x$

Step 4.2.7

By the Power Rule, the integral of $y^{- 4}$ with respect to $y$ is $- \frac{1}{3} y^{- 3}$ .

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

Step 4.2.8

Simplify.

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Step 4.2.8.1

Simplify.

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Step 4.2.8.1.1

Combine $y^{- 3}$ and $\frac{1}{3}$ .

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

Step 4.2.8.1.2

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2.8.2

Simplify.

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

Step 4.2.8.3

Simplify.

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Step 4.2.8.3.1

Multiply $- 1$ by $2$ .

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

Step 4.2.8.3.2

Combine $- 2$ and $\frac{1}{3 y^{3}}$ .

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

Step 4.2.8.3.3

Move the negative in front of the fraction.

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

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.

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

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Negate the exponent of $e^{x}$ and move it out of the denominator.

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

Step 4.3.2.2

Multiply the exponents in ${(e^{x})}^{- 1}$ .

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Move $- 1$ to the left of $x$ .

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

Step 4.3.2.2.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Simplify.

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Step 4.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.5.2

Multiply $\int e^{- x} d x$ by $1$ .

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

Step 4.3.6

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.6.1

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

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Step 4.3.6.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 4.3.6.1.2

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

$- \frac{d}{d x} [x]$

Step 4.3.6.1.3

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

$- 1 \cdot 1$

Step 4.3.6.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3.6.2

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

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

Step 4.3.7

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

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

Step 4.3.8

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

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

Step 4.3.9

Rewrite $- (x (- e^{- x}) - (e^{u} + C_{4}))$ as $- (- x e^{- x} - e^{u}) + C_{4}$ .

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

Step 4.3.10

Replace all occurrences of $u$ with $- x$ .

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

Step 4.3.11

Simplify.

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Step 4.3.11.1

Apply the distributive property.

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

Step 4.3.11.2

Multiply $- (- x e^{- x})$ .

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Step 4.3.11.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.11.2.2

Multiply $x$ by $1$ .

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

Step 4.3.11.3

Multiply $- - e^{- x}$ .

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Step 4.3.11.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.11.3.2

Multiply $e^{- x}$ by $1$ .

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

Step 4.3.12

Reorder terms.

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

Step 4.4

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

$- \frac{1}{y} - \frac{2}{3 y^{3}} = e^{- x} x + e^{- x} + K$