Calculus Examples

$x d x + \sec (x) \sin (y) d y = 0$

Step 1

Subtract $x d x$ from both sides of the equation.

$\sec (x) \sin (y) d y = - x d x$

Step 2

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

$\frac{1}{\sec (x)} (\sec (x) \sin (y)) d y = \frac{1}{\sec (x)} (- x) d x$

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $\sec (x)$ out of $\sec (x) \sin (y)$ .

$\frac{1}{\sec (x)} (\sec (x) (\sin (y))) d y = \frac{1}{\sec (x)} (- x) d x$

Step 3.1.2

Cancel the common factor.

$\frac{1}{\sec (x)} (\sec (x) \sin (y)) d y = \frac{1}{\sec (x)} (- x) d x$

Step 3.1.3

Rewrite the expression.

$\sin (y) d y = \frac{1}{\sec (x)} (- x) d x$

Step 3.2

Rewrite using the commutative property of multiplication.

$\sin (y) d y = - \frac{1}{\sec (x)} x d x$

Step 3.3

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

$\sin (y) d y = - \frac{1}{\frac{1}{\cos (x)}} x d x$

Step 3.4

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

$\sin (y) d y = - (1 \cos (x)) x d x$

Step 3.5

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

$\sin (y) d y = - \cos (x) x d x$

Step 3.6

Reorder factors in $- \cos (x) x$ .

$\sin (y) d y = - x \cos (x) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \sin (y) d y = \int - x \cos (x) d x$

Step 4.2

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

$- \cos (y) + C_{1} = \int - x \cos (x) d x$

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.

$- \cos (y) + C_{1} = - \int x \cos (x) d x$

Step 4.3.2

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

$- \cos (y) + C_{1} = - (x \sin (x) - \int \sin (x) d x)$

Step 4.3.3

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

$- \cos (y) + C_{1} = - (x \sin (x) - (- \cos (x) + C_{2}))$

Step 4.3.4

Rewrite $- (x \sin (x) - (- \cos (x) + C_{2}))$ as $- (x \sin (x) + \cos (x)) + C_{2}$ .

$- \cos (y) + C_{1} = - (x \sin (x) + \cos (x)) + C_{2}$

Step 4.4

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

$- \cos (y) = - (x \sin (x) + \cos (x)) + K$

Step 5

Solve for $y$ .

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Step 5.1

Rewrite the equation as $- (x \sin (x) + \cos (x)) + K = - \cos (y)$ .

$- (x \sin (x) + \cos (x)) + K = - \cos (y)$

Step 5.2

Apply the distributive property.

$- x \sin (x) - \cos (x) + K = - \cos (y)$

Step 5.3

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 5.3.1

Add $x \sin (x)$ to both sides of the equation.

$- \cos (x) + K = - \cos (y) + x \sin (x)$

Step 5.3.2

Subtract $K$ from both sides of the equation.

$- \cos (x) = - \cos (y) + x \sin (x) - K$

Step 5.4

Divide each term in $- \cos (x) = - \cos (y) + x \sin (x) - K$ by $- 1$ and simplify.

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Step 5.4.1

Divide each term in $- \cos (x) = - \cos (y) + x \sin (x) - K$ by $- 1$ .

$\frac{- \cos (x)}{- 1} = \frac{- \cos (y)}{- 1} + \frac{x \sin (x)}{- 1} + \frac{- K}{- 1}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (x)}{1} = \frac{- \cos (y)}{- 1} + \frac{x \sin (x)}{- 1} + \frac{- K}{- 1}$

Step 5.4.2.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{- \cos (y)}{- 1} + \frac{x \sin (x)}{- 1} + \frac{- K}{- 1}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Dividing two negative values results in a positive value.

$\cos (x) = \frac{\cos (y)}{1} + \frac{x \sin (x)}{- 1} + \frac{- K}{- 1}$

Step 5.4.3.1.2

Divide $\cos (y)$ by $1$ .

$\cos (x) = \cos (y) + \frac{x \sin (x)}{- 1} + \frac{- K}{- 1}$

Step 5.4.3.1.3

Move the negative one from the denominator of $\frac{x \sin (x)}{- 1}$ .

$\cos (x) = \cos (y) - 1 \cdot (x \sin (x)) + \frac{- K}{- 1}$

Step 5.4.3.1.4

Rewrite $- 1 \cdot (x \sin (x))$ as $- (x \sin (x))$ .

$\cos (x) = \cos (y) - (x \sin (x)) + \frac{- K}{- 1}$

Step 5.4.3.1.5

Dividing two negative values results in a positive value.

$\cos (x) = \cos (y) - x \sin (x) + \frac{K}{1}$

Step 5.4.3.1.6

Divide $K$ by $1$ .

$\cos (x) = \cos (y) - x \sin (x) + K$

Step 5.5

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$x = \arccos (\cos (y) - x \sin (x) + K)$

Step 5.6

Rewrite the equation as $\arccos (\cos (y) - x \sin (x) + K) = x$ .

$\arccos (\cos (y) - x \sin (x) + K) = x$

Step 5.7

Take the inverse arccosine of both sides of the equation to extract $\cos (y)$ from inside the arccosine.

$\cos (y) - x \sin (x) + K = \cos (x)$

Step 5.8

Move all terms not containing $\cos (y)$ to the right side of the equation.

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Step 5.8.1

Add $x \sin (x)$ to both sides of the equation.

$\cos (y) + K = \cos (x) + x \sin (x)$

Step 5.8.2

Subtract $K$ from both sides of the equation.

$\cos (y) = \cos (x) + x \sin (x) - K$

Step 5.9

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$y = \arccos (\cos (x) + x \sin (x) - K)$

Step 5.10

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\arccos (\cos (y) - x \sin (x) + K) = x$

Step 5.11

Take the inverse arccosine of both sides of the equation to extract $\cos (y)$ from inside the arccosine.

$\cos (y) - x \sin (x) + K = \cos (x)$

Step 5.12

Move all terms not containing $\cos (y)$ to the right side of the equation.

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Step 5.12.1

Add $x \sin (x)$ to both sides of the equation.

$\cos (y) + K = \cos (x) + x \sin (x)$

Step 5.12.2

Subtract $K$ from both sides of the equation.

$\cos (y) = \cos (x) + x \sin (x) - K$

Step 5.13

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$y = \arccos (\cos (x) + x \sin (x) - K)$

Step 6

Simplify the constant of integration.

$y = \arccos (\cos (x) + x \sin (x) + K)$