Calculus Examples

$\frac{d y}{d x} = 7 \cos (x) \sec (y)$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{\sec (y)} \frac{d y}{d x} = \frac{1}{\sec (y)} (7 \cos (x) \sec (y))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 \frac{1}{\sec (y)} \cos (x) \sec (y)$

Step 1.2.2

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

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 \frac{1}{\frac{1}{\cos (y)}} \cos (x) \sec (y)$

Step 1.2.3

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

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 (1 \cos (y)) \cos (x) \sec (y)$

Step 1.2.4

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

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 \cos (y) \cos (x) \sec (y)$

Step 1.2.5

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

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 \cos (y) \cos (x) \frac{1}{\cos (y)}$

Step 1.2.6

Cancel the common factor of $\cos (y)$ .

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Step 1.2.6.1

Factor $\cos (y)$ out of $7 \cos (y) \cos (x)$ .

$\frac{1}{\sec (y)} \frac{d y}{d x} = \cos (y) (7 \cos (x)) \frac{1}{\cos (y)}$

Step 1.2.6.2

Cancel the common factor.

$\frac{1}{\sec (y)} \frac{d y}{d x} = \cos (y) (7 \cos (x)) \frac{1}{\cos (y)}$

Step 1.2.6.3

Rewrite the expression.

$\frac{1}{\sec (y)} \frac{d y}{d x} = 7 \cos (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{\sec (y)} d y = 7 \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sec (y)} d y = \int 7 \cos (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

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

$\int \frac{1}{\frac{1}{\cos (y)}} d y = \int 7 \cos (x) d x$

Step 2.2.1.2

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

$\int 1 \cos (y) d y = \int 7 \cos (x) d x$

Step 2.2.1.3

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

$\int \cos (y) d y = \int 7 \cos (x) d x$

Step 2.2.2

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

$\sin (y) + C_{1} = \int 7 \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\sin (y) + C_{1} = 7 \int \cos (x) d x$

Step 2.3.2

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

$\sin (y) + C_{1} = 7 (\sin (x) + C_{2})$

Step 2.3.3

Simplify.

$\sin (y) + C_{1} = 7 \sin (x) + C_{2}$

Step 2.4

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

$\sin (y) = 7 \sin (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $7 \sin (x) + K = \sin (y)$ .

$7 \sin (x) + K = \sin (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$7 \sin (x) = \sin (y) - K$

Step 3.3

Divide each term in $7 \sin (x) = \sin (y) - K$ by $7$ and simplify.

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Step 3.3.1

Divide each term in $7 \sin (x) = \sin (y) - K$ by $7$ .

$\frac{7 \sin (x)}{7} = \frac{\sin (y)}{7} + \frac{- K}{7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{7 \sin (x)}{7} = \frac{\sin (y)}{7} + \frac{- K}{7}$

Step 3.3.2.1.2

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

$\sin (x) = \frac{\sin (y)}{7} + \frac{- K}{7}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$\sin (x) = \frac{\sin (y)}{7} - \frac{K}{7}$

Step 3.4

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

$x = \arcsin (\frac{\sin (y)}{7} - \frac{K}{7})$

Step 3.5

Rewrite the equation as $\arcsin (\frac{\sin (y)}{7} - \frac{K}{7}) = x$ .

$\arcsin (\frac{\sin (y)}{7} - \frac{K}{7}) = x$

Step 3.6

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

$\frac{\sin (y)}{7} - \frac{K}{7} = \sin (x)$

Step 3.7

Add $\frac{K}{7}$ to both sides of the equation.

$\frac{\sin (y)}{7} = \sin (x) + \frac{K}{7}$

Step 3.8

Multiply both sides of the equation by $7$ .

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.9

Simplify both sides of the equation.

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Step 3.9.1

Simplify the left side.

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Step 3.9.1.1

Cancel the common factor of $7$ .

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Step 3.9.1.1.1

Cancel the common factor.

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.9.1.1.2

Rewrite the expression.

$\sin (y) = 7 (\sin (x) + \frac{K}{7})$

Step 3.9.2

Simplify the right side.

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Step 3.9.2.1

Simplify $7 (\sin (x) + \frac{K}{7})$ .

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Step 3.9.2.1.1

Apply the distributive property.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.9.2.1.2

Cancel the common factor of $7$ .

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Step 3.9.2.1.2.1

Cancel the common factor.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.9.2.1.2.2

Rewrite the expression.

$\sin (y) = 7 \sin (x) + K$

Step 3.10

Reorder $7 \sin (x)$ and $K$ .

$\sin (y) = K + 7 \sin (x)$

Step 3.11

Add $\frac{K}{7}$ to both sides of the equation.

$\frac{\sin (y)}{7} = \sin (x) + \frac{K}{7}$

Step 3.12

Multiply both sides of the equation by $7$ .

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.13

Simplify both sides of the equation.

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Step 3.13.1

Simplify the left side.

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Step 3.13.1.1

Cancel the common factor of $7$ .

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Step 3.13.1.1.1

Cancel the common factor.

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.13.1.1.2

Rewrite the expression.

$\sin (y) = 7 (\sin (x) + \frac{K}{7})$

Step 3.13.2

Simplify the right side.

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Step 3.13.2.1

Simplify $7 (\sin (x) + \frac{K}{7})$ .

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Step 3.13.2.1.1

Apply the distributive property.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.13.2.1.2

Cancel the common factor of $7$ .

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Step 3.13.2.1.2.1

Cancel the common factor.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.13.2.1.2.2

Rewrite the expression.

$\sin (y) = 7 \sin (x) + K$

Step 3.14

Reorder $7 \sin (x)$ and $K$ .

$\sin (y) = K + 7 \sin (x)$

Step 3.15

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

$y = \arcsin (K + 7 \sin (x))$

Step 3.16

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

$\arcsin (\frac{\sin (y)}{7} - \frac{K}{7}) = x$

Step 3.17

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

$\frac{\sin (y)}{7} - \frac{K}{7} = \sin (x)$

Step 3.18

Add $\frac{K}{7}$ to both sides of the equation.

$\frac{\sin (y)}{7} = \sin (x) + \frac{K}{7}$

Step 3.19

Multiply both sides of the equation by $7$ .

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.20

Simplify both sides of the equation.

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Step 3.20.1

Simplify the left side.

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Step 3.20.1.1

Cancel the common factor of $7$ .

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Step 3.20.1.1.1

Cancel the common factor.

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.20.1.1.2

Rewrite the expression.

$\sin (y) = 7 (\sin (x) + \frac{K}{7})$

Step 3.20.2

Simplify the right side.

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Step 3.20.2.1

Simplify $7 (\sin (x) + \frac{K}{7})$ .

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Step 3.20.2.1.1

Apply the distributive property.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.20.2.1.2

Cancel the common factor of $7$ .

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Step 3.20.2.1.2.1

Cancel the common factor.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.20.2.1.2.2

Rewrite the expression.

$\sin (y) = 7 \sin (x) + K$

Step 3.21

Reorder $7 \sin (x)$ and $K$ .

$\sin (y) = K + 7 \sin (x)$

Step 3.22

Add $\frac{K}{7}$ to both sides of the equation.

$\frac{\sin (y)}{7} = \sin (x) + \frac{K}{7}$

Step 3.23

Multiply both sides of the equation by $7$ .

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.24

Simplify both sides of the equation.

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Step 3.24.1

Simplify the left side.

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Step 3.24.1.1

Cancel the common factor of $7$ .

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Step 3.24.1.1.1

Cancel the common factor.

$7 \frac{\sin (y)}{7} = 7 (\sin (x) + \frac{K}{7})$

Step 3.24.1.1.2

Rewrite the expression.

$\sin (y) = 7 (\sin (x) + \frac{K}{7})$

Step 3.24.2

Simplify the right side.

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Step 3.24.2.1

Simplify $7 (\sin (x) + \frac{K}{7})$ .

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Step 3.24.2.1.1

Apply the distributive property.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.24.2.1.2

Cancel the common factor of $7$ .

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Step 3.24.2.1.2.1

Cancel the common factor.

$\sin (y) = 7 \sin (x) + 7 \frac{K}{7}$

Step 3.24.2.1.2.2

Rewrite the expression.

$\sin (y) = 7 \sin (x) + K$

Step 3.25

Reorder $7 \sin (x)$ and $K$ .

$\sin (y) = K + 7 \sin (x)$

Step 3.26

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

$y = \arcsin (K + 7 \sin (x))$