English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

solvefor y,integral of x=csc^2(y/2)

Solution

solve for

y, \int x d x = csc^{2} (\frac{y}{2})

Solution

y = 2 arccsc (\int x d x) + 4 πn, y = 2 π + 2 arccsc (- \int x d x) + 4 πn, y = 2 arccsc (- \int x d x) + 4 πn, y = 2 π + 2 arccsc (\int x d x) + 4 πn

Solution steps

\int x d x = csc^{2} (\frac{y}{2})

Switch sides

csc^{2} (\frac{y}{2}) = \int x d x

Solve by substitution

csc^{2} (\frac{y}{2}) = \int x d x

Let:

csc (\frac{y}{2}) = u

u^{2} = \int x d x

u^{2} = \int x d x : u = \int x d x, u = - \int x d x

u^{2} = \int x d x

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \int x d x, u = - \int x d x

Substitute back

u = csc (\frac{y}{2})

csc (\frac{y}{2}) = \int x d x, csc (\frac{y}{2}) = - \int x d x

csc (\frac{y}{2}) = \int x d x, csc (\frac{y}{2}) = - \int x d x

csc (\frac{y}{2}) = \int x d x : y = 2 arccsc (\int x d x) + 4 πn, y = 2 π + 2 arccsc (- \int x d x) + 4 πn

csc (\frac{y}{2}) = \int x d x

Apply trig inverse properties

csc (\frac{y}{2}) = \int x d x

General solutions for

csc (\frac{y}{2}) = \int x d x

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π + arccsc (a) + 2 πn

\frac{y}{2} = arccsc (\int x d x) + 2 πn, \frac{y}{2} = π + arccsc (- \int x d x) + 2 πn

\frac{y}{2} = arccsc (\int x d x) + 2 πn, \frac{y}{2} = π + arccsc (- \int x d x) + 2 πn

Solve

\frac{y}{2} = arccsc (\int x d x) + 2 πn : y = 2 arccsc (\int x d x) + 4 πn

\frac{y}{2} = arccsc (\int x d x) + 2 πn

Multiply both sides by

2

\frac{y}{2} = arccsc (\int x d x) + 2 πn

Multiply both sides by

2

\frac{2 y}{2} = 2 arccsc (\int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} = 2 arccsc (\int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

2 arccsc (\int x d x) + 2 \cdot 2 πn : 2 arccsc (\int x d x) + 4 πn

2 arccsc (\int x d x) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccsc (\int x d x) + 4 πn

y = 2 arccsc (\int x d x) + 4 πn

y = 2 arccsc (\int x d x) + 4 πn

y = 2 arccsc (\int x d x) + 4 πn

Solve

\frac{y}{2} = π + arccsc (- \int x d x) + 2 πn : y = 2 π + 2 arccsc (- \int x d x) + 4 πn

\frac{y}{2} = π + arccsc (- \int x d x) + 2 πn

Multiply both sides by

2

\frac{y}{2} = π + arccsc (- \int x d x) + 2 πn

Multiply both sides by

2

\frac{2 y}{2} = 2 π + 2 arccsc (- \int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} = 2 π + 2 arccsc (- \int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

2 π + 2 arccsc (- \int x d x) + 2 \cdot 2 πn : 2 π + 2 arccsc (- \int x d x) + 4 πn

2 π + 2 arccsc (- \int x d x) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arccsc (- \int x d x) + 4 πn

y = 2 π + 2 arccsc (- \int x d x) + 4 πn

y = 2 π + 2 arccsc (- \int x d x) + 4 πn

y = 2 π + 2 arccsc (- \int x d x) + 4 πn

y = 2 arccsc (\int x d x) + 4 πn, y = 2 π + 2 arccsc (- \int x d x) + 4 πn

csc (\frac{y}{2}) = - \int x d x : y = 2 arccsc (- \int x d x) + 4 πn, y = 2 π + 2 arccsc (\int x d x) + 4 πn

csc (\frac{y}{2}) = - \int x d x

Apply trig inverse properties

csc (\frac{y}{2}) = - \int x d x

General solutions for

csc (\frac{y}{2}) = - \int x d x

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π + arccsc (a) + 2 πn

\frac{y}{2} = arccsc (- \int x d x) + 2 πn, \frac{y}{2} = π + arccsc (\int x d x) + 2 πn

\frac{y}{2} = arccsc (- \int x d x) + 2 πn, \frac{y}{2} = π + arccsc (\int x d x) + 2 πn

Solve

\frac{y}{2} = arccsc (- \int x d x) + 2 πn : y = 2 arccsc (- \int x d x) + 4 πn

\frac{y}{2} = arccsc (- \int x d x) + 2 πn

Multiply both sides by

2

\frac{y}{2} = arccsc (- \int x d x) + 2 πn

Multiply both sides by

2

\frac{2 y}{2} = 2 arccsc (- \int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} = 2 arccsc (- \int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

2 arccsc (- \int x d x) + 2 \cdot 2 πn : 2 arccsc (- \int x d x) + 4 πn

2 arccsc (- \int x d x) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccsc (- \int x d x) + 4 πn

y = 2 arccsc (- \int x d x) + 4 πn

y = 2 arccsc (- \int x d x) + 4 πn

y = 2 arccsc (- \int x d x) + 4 πn

Solve

\frac{y}{2} = π + arccsc (\int x d x) + 2 πn : y = 2 π + 2 arccsc (\int x d x) + 4 πn

\frac{y}{2} = π + arccsc (\int x d x) + 2 πn

Multiply both sides by

2

\frac{y}{2} = π + arccsc (\int x d x) + 2 πn

Multiply both sides by

2

\frac{2 y}{2} = 2 π + 2 arccsc (\int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} = 2 π + 2 arccsc (\int x d x) + 2 \cdot 2 πn

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

2 π + 2 arccsc (\int x d x) + 2 \cdot 2 πn : 2 π + 2 arccsc (\int x d x) + 4 πn

2 π + 2 arccsc (\int x d x) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arccsc (\int x d x) + 4 πn

y = 2 π + 2 arccsc (\int x d x) + 4 πn

y = 2 π + 2 arccsc (\int x d x) + 4 πn

y = 2 π + 2 arccsc (\int x d x) + 4 πn

y = 2 arccsc (- \int x d x) + 4 πn, y = 2 π + 2 arccsc (\int x d x) + 4 πn

Combine all the solutions

y = 2 arccsc (\int x d x) + 4 πn, y = 2 π + 2 arccsc (- \int x d x) + 4 πn, y = 2 arccsc (- \int x d x) + 4 πn, y = 2 π + 2 arccsc (\int x d x) + 4 πn

Popular Examples

sec(x)+sqrt(2)=0,-pi<= x<= (3pi)/2

sec (x) + 2 = 0, - π \leq x \leq \frac{3 π}{2}

sin^2(x)= 1/9

sin^{2} (x) = \frac{1}{9}

2sin^2(x)+sin(x)= 1/2

2 sin^{2} (x) + sin (x) = \frac{1}{2}

tan(x)=0.327

tan (x) = 0.327

(cos(2x))/(sin(x))=1

\frac{cos ( 2 x )}{sin ( x )} = 1