What is the integral of csc(θ)(csc(θ)+cot(θ)) ?

The integral of csc(θ)(csc(θ)+cot(θ)) is -cot(θ/2)+C

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integral of csc(θ)(csc(θ)+cot(θ))

\int csc (θ) (csc (θ) + cot (θ)) d θ

- cot (\frac{θ}{2}) + C

Solution steps

\int csc (θ) (csc (θ) + cot (θ)) d θ

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{u}

Substitute back

u = tan (\frac{θ}{2})

= - \frac{1}{tan ( \frac{θ}{2} )}

Use the basic trigonometric identity:

\frac{1}{tan ( x )} = cot (x)

= - cot (\frac{θ}{2})

Add a constant to the solution

= - cot (\frac{θ}{2}) + C

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What is the integral of csc(θ)(csc(θ)+cot(θ)) ?
The integral of csc(θ)(csc(θ)+cot(θ)) is -cot(θ/2)+C