What is the integral of 1/(csc(x)+cot(x)) ?

The integral of 1/(csc(x)+cot(x)) is ln(sec^2(x/2))+C

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integral of 1/(csc(x)+cot(x))

\int \frac{1}{csc ( x ) + cot ( x )} d x

ln (sec^{2} (\frac{x}{2})) + C

Solution steps

\int \frac{1}{csc ( x ) + cot ( x )} d x

Apply u-substitution

: \int \frac{2 u}{1 + u ^{2}} d u

= \int \frac{2 u}{1 + u ^{2}} d u

Take the constant out:

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

= 2 \cdot \int \frac{u}{1 + u ^{2}} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 2 \cdot \frac{1}{2} ln 1 + tan^{2} (\frac{x}{2})

Simplify

2 \cdot \frac{1}{2} ln 1 + tan^{2} (\frac{x}{2}) : ln (sec^{2} (\frac{x}{2}))

= ln (sec^{2} (\frac{x}{2}))

Add a constant to the solution

= ln (sec^{2} (\frac{x}{2})) + C

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What is the integral of 1/(csc(x)+cot(x)) ?
The integral of 1/(csc(x)+cot(x)) is ln(sec^2(x/2))+C