Evaluate the indefinite integral `int (csc(t)cot(t))/(1+9csc^2(t))dt`

`int (csc(t) cot(t) ) / (1+9csc^2(t)) dt`

`= int (csc(t)cot(t))/(1+(3csct)^2) dt`

Note that the derivative of co-secant function is (csc u)' = -csc u cot u. Since the numerator of the integrand is similar to the of the derivative of co-secant, use u-substitution method to integrate.

So let,

`u = 3csc(t)`

Then, differentiate u.

`du = -3csc(t)cot(t) dt`

`-(du)/3=csc(t)cot(t) dt`

And replace the t variable of the integrand with u.

`int 1/(1+u^2)*(- (du)/3)`

`=-1/3 int 1/(1+u^2) du`

Apply the formula `int 1/(a^2 + u^2)du = 1/a tan^(-1) u/a+C` .

`= -1/3 tan^(-1) u + C`

Substitute back u=3csc(t) to return to the original variable.

`=-1/3 tan^(-1) (3csc (t) ) + C`

Hence, `int (csc(t)cot(t))/(1+9csc^2t)dt = -1/3 tan^(-1) (3csc(t))+C` .