Calculus Examples

$\frac{d y}{d x} = \cos (x) \cos^{2} (y)$ , $y (0) = \frac{π}{4}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\cos^{2} (y)}$ .

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (\cos (x) \cos^{2} (y))$

Step 1.2

Cancel the common factor of $\cos^{2} (y)$ .

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Step 1.2.1

Factor $\cos^{2} (y)$ out of $\cos (x) \cos^{2} (y)$ .

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (\cos^{2} (y) \cos (x))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \frac{1}{\cos^{2} (y)} (\cos^{2} (y) \cos (x))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{\cos^{2} (y)} \frac{d y}{d x} = \cos (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{\cos^{2} (y)} d y = \cos (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\cos^{2} (y)} d y = \int \cos (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Convert from $\frac{1}{\cos^{2} (y)}$ to $\sec^{2} (y)$ .

$\int \sec^{2} (y) d y = \int \cos (x) d x$

Step 2.2.2

Since the derivative of $\tan (y)$ is $\sec^{2} (y)$ , the integral of $\sec^{2} (y)$ is $\tan (y)$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Subtract $K$ from both sides of the equation.

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

Step 3.3

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

$x = \arcsin (\tan (y) - K)$

Step 3.4

Rewrite the equation as $\arcsin (\tan (y) - K) = x$ .

$\arcsin (\tan (y) - K) = x$

Step 3.5

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

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

Step 3.6

Add $K$ to both sides of the equation.

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

Step 3.7

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

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

Step 3.8

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

$\arcsin (\tan (y) - K) = x$

Step 3.9

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

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

Step 3.10

Add $K$ to both sides of the equation.

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

Step 3.11

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

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

Step 4

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $\frac{π}{4}$ for $y$ in $y = \arctan (\sin (x) + K)$ .

$\frac{π}{4} = \arctan (\sin (0) + K)$

Step 5

Solve for $K$ .

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Step 5.1

Rewrite the equation as $\arctan (\sin (0) + K) = \frac{π}{4}$ .

$\arctan (\sin (0) + K) = \frac{π}{4}$

Step 5.2

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

$\sin (0) + K = \arcsin (\frac{π}{4})$

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $\sin (0) + K$ .

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Step 5.3.1.1

The exact value of $\sin (0)$ is $0$ .

$0 + K = \arcsin (\frac{π}{4})$

Step 5.3.1.2

Add $0$ and $K$ .

$K = \arcsin (\frac{π}{4})$

Step 5.4

Simplify the right side.

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Step 5.4.1

Evaluate $\arcsin (\frac{π}{4})$ .

$K = 0.90333911$

Step 5.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$K = (3.14159265) - 0.90333911$

Step 5.6

Solve for $K$ .

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Step 5.6.1

Remove parentheses.

$K = 3.14159265 - 0.90333911$

Step 5.6.2

Remove parentheses.

$K = (3.14159265) - 0.90333911$

Step 5.6.3

Subtract $0.90333911$ from $3.14159265$ .

$K = 2.23825354$

Step 5.7

Exclude the solutions that do not make $\frac{π}{4} = \arctan (\sin (0) + K)$ true.

$No solution$