Calculus Examples

$x^{5} y d x + ((y^{4} + 3 y^{2}) \csc (x)) d y = 0$

Step 1

Subtract $x^{5} y d x$ from both sides of the equation.

$(y^{4} + 3 y^{2}) \csc (x) d y = - x^{5} y d x$

Step 2

Multiply both sides by $\frac{1}{\csc (x) y}$ .

$\frac{1}{\csc (x) y} ((y^{4} + 3 y^{2}) \csc (x)) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $\csc (x)$ .

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Step 3.1.1

Factor $\csc (x)$ out of $\csc (x) y$ .

$\frac{1}{\csc (x) (y)} ((y^{4} + 3 y^{2}) \csc (x)) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.1.2

Factor $\csc (x)$ out of $(y^{4} + 3 y^{2}) \csc (x)$ .

$\frac{1}{\csc (x) (y)} (\csc (x) (y^{4} + 3 y^{2})) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.1.3

Cancel the common factor.

$\frac{1}{\csc (x) y} (\csc (x) (y^{4} + 3 y^{2})) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.1.4

Rewrite the expression.

$\frac{1}{y} (y^{4} + 3 y^{2}) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.2

Multiply $\frac{1}{y}$ by $y^{4} + 3 y^{2}$ .

$\frac{y^{4} + 3 y^{2}}{y} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.3

Factor $y^{2}$ out of $y^{4} + 3 y^{2}$ .

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Step 3.3.1

Factor $y^{2}$ out of $y^{4}$ .

$\frac{y^{2} y^{2} + 3 y^{2}}{y} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.3.2

Factor $y^{2}$ out of $3 y^{2}$ .

$\frac{y^{2} y^{2} + y^{2} \cdot 3}{y} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.3.3

Factor $y^{2}$ out of $y^{2} y^{2} + y^{2} \cdot 3$ .

$\frac{y^{2} (y^{2} + 3)}{y} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4

Cancel the common factor of $y^{2}$ and $y$ .

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Step 3.4.1

Factor $y$ out of $y^{2} (y^{2} + 3)$ .

$\frac{y (y (y^{2} + 3))}{y} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Raise $y$ to the power of $1$ .

$\frac{y (y (y^{2} + 3))}{y^{1}} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4.2.2

Factor $y$ out of $y^{1}$ .

$\frac{y (y (y^{2} + 3))}{y \cdot 1} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4.2.3

Cancel the common factor.

$\frac{y (y (y^{2} + 3))}{y \cdot 1} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4.2.4

Rewrite the expression.

$\frac{y (y^{2} + 3)}{1} d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.4.2.5

Divide $y (y^{2} + 3)$ by $1$ .

$y (y^{2} + 3) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.5

Apply the distributive property.

$(y \cdot y^{2} + y \cdot 3) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.6

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

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Step 3.6.1

Multiply $y$ by $y^{2}$ .

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Step 3.6.1.1

Raise $y$ to the power of $1$ .

$(y^{1} y^{2} + y \cdot 3) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.6.1.2

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

$(y^{1 + 2} + y \cdot 3) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.6.2

Add $1$ and $2$ .

$(y^{3} + y \cdot 3) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.7

Move $3$ to the left of $y$ .

$(y^{3} + 3 y) d y = \frac{1}{\csc (x) y} (- x^{5} y) d x$

Step 3.8

Rewrite using the commutative property of multiplication.

$(y^{3} + 3 y) d y = - \frac{1}{\csc (x) y} (x^{5} y) d x$

Step 3.9

Cancel the common factor of $y$ .

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Step 3.9.1

Move the leading negative in $- \frac{1}{\csc (x) y}$ into the numerator.

$(y^{3} + 3 y) d y = \frac{- 1}{\csc (x) y} (x^{5} y) d x$

Step 3.9.2

Factor $y$ out of $\csc (x) y$ .

$(y^{3} + 3 y) d y = \frac{- 1}{y \csc (x)} (x^{5} y) d x$

Step 3.9.3

Factor $y$ out of $x^{5} y$ .

$(y^{3} + 3 y) d y = \frac{- 1}{y \csc (x)} (y x^{5}) d x$

Step 3.9.4

Cancel the common factor.

$(y^{3} + 3 y) d y = \frac{- 1}{y \csc (x)} (y x^{5}) d x$

Step 3.9.5

Rewrite the expression.

$(y^{3} + 3 y) d y = \frac{- 1}{\csc (x)} x^{5} d x$

Step 3.10

Combine $\frac{- 1}{\csc (x)}$ and $x^{5}$ .

$(y^{3} + 3 y) d y = \frac{- x^{5}}{\csc (x)} d x$

Step 3.11

Move the negative in front of the fraction.

$(y^{3} + 3 y) d y = - \frac{x^{5}}{\csc (x)} d x$

Step 3.12

Separate fractions.

$(y^{3} + 3 y) d y = - (\frac{x^{5}}{1} \cdot \frac{1}{\csc (x)}) d x$

Step 3.13

Rewrite $\csc (x)$ in terms of sines and cosines.

$(y^{3} + 3 y) d y = - (\frac{x^{5}}{1} \cdot \frac{1}{\frac{1}{\sin (x)}}) d x$

Step 3.14

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$(y^{3} + 3 y) d y = - (\frac{x^{5}}{1} (1 \sin (x))) d x$

Step 3.15

Multiply $\sin (x)$ by $1$ .

$(y^{3} + 3 y) d y = - (\frac{x^{5}}{1} \sin (x)) d x$

Step 3.16

Divide $x^{5}$ by $1$ .

$(y^{3} + 3 y) d y = - x^{5} \sin (x) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int y^{3} + 3 y d y = \int - x^{5} \sin (x) d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the single integral into multiple integrals.

$\int y^{3} d y + \int 3 y d y = \int - x^{5} \sin (x) d x$

Step 4.2.2

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

$\frac{1}{4} y^{4} + C_{1} + \int 3 y d y = \int - x^{5} \sin (x) d x$

Step 4.2.3

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$\frac{1}{4} y^{4} + C_{1} + 3 \int y d y = \int - x^{5} \sin (x) d x$

Step 4.2.4

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

$\frac{1}{4} y^{4} + C_{1} + 3 (\frac{1}{2} y^{2} + C_{2}) = \int - x^{5} \sin (x) d x$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Simplify.

$\frac{1}{4} y^{4} + 3 (\frac{1}{2}) y^{2} + C_{3} = \int - x^{5} \sin (x) d x$

Step 4.2.5.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = \int - x^{5} \sin (x) d x$

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.

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - \int x^{5} \sin (x) d x$

Step 4.3.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{5}$ and $d v = \sin (x)$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) - \int - \cos (x) (5 x^{4}) d x)$

Step 4.3.3

Multiply $5$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) - \int - 5 \cos (x) x^{4} d x)$

Step 4.3.4

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) - (- 5 \int \cos (x) x^{4} d x))$

Step 4.3.5

Multiply $- 5$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 \int \cos (x) x^{4} d x)$

Step 4.3.6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{4}$ and $d v = \cos (x)$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - \int \sin (x) (4 x^{3}) d x))$

Step 4.3.7

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - (4 \int \sin (x) (x^{3}) d x)))$

Step 4.3.8

Multiply $4$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 \int \sin (x) (x^{3}) d x))$

Step 4.3.9

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = \sin (x)$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) - \int - \cos (x) (3 x^{2}) d x)))$

Step 4.3.10

Multiply $3$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) - \int - 3 \cos (x) x^{2} d x)))$

Step 4.3.11

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) - (- 3 \int \cos (x) x^{2} d x))))$

Step 4.3.12

Multiply $- 3$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 \int \cos (x) x^{2} d x)))$

Step 4.3.13

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \cos (x)$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - \int \sin (x) (2 x) d x))))$

Step 4.3.14

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

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - (2 \int \sin (x) (x) d x)))))$

Step 4.3.15

Multiply $2$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 \int \sin (x) (x) d x))))$

Step 4.3.16

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

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) - \int - \cos (x) d x)))))$

Step 4.3.17

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

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) - - \int \cos (x) d x)))))$

Step 4.3.18

Simplify.

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Step 4.3.18.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + 1 \int \cos (x) d x)))))$

Step 4.3.18.2

Multiply $\int \cos (x) d x$ by $1$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \int \cos (x) d x)))))$

Step 4.3.19

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

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x) + C_{4})))))$

Step 4.3.20

Rewrite $- (x^{5} (- \cos (x)) + 5 (x^{4} \sin (x) - 4 (x^{3} (- \cos (x)) + 3 (x^{2} \sin (x) - 2 (x (- \cos (x)) + \sin (x) + C_{4})))))$ as $- (- x^{5} \cos (x) + 5 x^{4} \sin (x) + 20 x^{3} \cos (x) - 60 x^{2} \sin (x) - 120 x \cos (x) + 120 \sin (x)) + C_{4}$ .

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} + C_{3} = - (- x^{5} \cos (x) + 5 x^{4} \sin (x) + 20 x^{3} \cos (x) - 60 x^{2} \sin (x) - 120 x \cos (x) + 120 \sin (x)) + C_{4}$

Step 4.4

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

$\frac{1}{4} y^{4} + \frac{3}{2} y^{2} = - (- x^{5} \cos (x) + 5 x^{4} \sin (x) + 20 x^{3} \cos (x) - 60 x^{2} \sin (x) - 120 x \cos (x) + 120 \sin (x)) + K$