Compute ∫ ((a cos(x))/(b sin(x) +c))dx. in terms of a,b, and c.

Question

Compute ∫ ((a cos(x))/(b sin(x) +c))dx. in terms of a,b, and c. You may assume "b" does not equal zero. Use "+C" as a constant to indicate a family of solutions differing by a constant.

∫ ((a cos(x))/(b sin(x) +c))dx = ???

Answer 1

Your original attempt was very close! I believe you just dropped a constant somewhere in your work.

For this problem, let's use a u-substitution to make it easier to solve. If we set u = bsin(x)+c, we can differentiate it with respect to x to find du = bcos(x)dx.

Now, we must adjust our original integral to be in terms of u. We can write ∫(a/b)(1/(bsin(x)+c))(bcos(x)dx) ---> ∫(a/b)(1/u)du = (a/b)ln(u)+C (*note that the +C at the end is a capital C, not the lowercase c from the original problem).

Finally, let's replace u to get our overall solution: (a/b)ln(bsin(x)+c)+C