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

The integral of (csc(x)cot(x))/(1+csc^2(x)) is arctan(sin(x))+C

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integral of (csc(x)cot(x))/(1+csc^2(x))

\int \frac{csc ( x ) cot ( x )}{1 + csc ^{2} ( x )} d x

arctan (sin (x)) + C

Solution steps

\int \frac{csc ( x ) cot ( x )}{1 + csc ^{2} ( x )} d x

Apply Trigonometric Substitution

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

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

Use the common integral:

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

= arctan (u)

Substitute back

u = sin (x)

= arctan (sin (x))

Add a constant to the solution

= arctan (sin (x)) + C

\frac{d}{d x} (tan (x) + x sec^{2} (x))

sin (tan (x) + sec (x))

\frac{d}{d x} (cos (5 x - \frac{π}{2}))

\frac{s}{s ^{2} - s + 1}

\frac{3}{s ( s + 3 )}

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