What is the integral of csc(x/2) ?

The integral of csc(x/2) is 2ln|tan(x/4)|+C

Popular >

integral of csc(x/2)

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

2 ln tan (\frac{x}{4}) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{sin ( \frac{x}{2} )} d x

Apply u-substitution

: \int 2 csc (u) d u

= \int 2 csc (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int csc (u) d u

Apply u-substitution

: \int \frac{1}{v} d v

= 2 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 ln ∣ v ∣

Substitute back

= 2 ln tan (\frac{\frac{x}{2}}{2})

\frac{\frac{x}{2}}{2} = \frac{x}{4}

= 2 ln tan (\frac{x}{4})

Add a constant to the solution

= 2 ln tan (\frac{x}{4}) + C

\frac{\partial}{\partial y} (10 x y)

\frac{1}{x ^{2} - 2 x - 3}

x \to 4 lim (\frac{1}{x + 4})

\int sin^{11} (x) cos^{5} (x) d x

\int (\frac{19.8}{x} - 9 x^{2}) d x