Calculus of a Single Variable, Chapter 5, 5.2, Section 5.2, Problem 38

int (csc^2(t))/(cot(t))dt=
We will use the following formula:
int (f'(x))/(f(x))dx=ln|f(x)|+C.
We will use the following formula:
The formula tells us that if we have integral of rational function where numerator is equal to derivative of denominator, then the integral is equal to natural logarithm of the denominator plus some constant. The proof of the formula can be obtained by simply integrating the right-hand side.
Since (cot(t))'=-csc^2(t) we first have to modify our integral in order to apply the formula. We will put the minus sign in the numerator but then we must also put minus sign in front of the integral (-1 times-1=1).
-int(-csc^2(t))/cot(t)dt=
Now we apply the formula.
-ln(cot(t))+C=
We can further simplify the result by using rule for logarithm of power: r log_b x=log_b x^r
ln(cot^(-1)(t))+C=
And since cot^(-1)(t)=tan(t) we have
ln(tan(t))+C