integral(sin(2x)+5tan^2 x csc^2 x)dx=

To solve the integral ∫(sin(2x) + 5tan^2x * csc^2x) dx, we can break it down into two separate integrals:

∫sin(2x) dx + ∫(5tan^2x * csc^2x) dx

Let's solve them one by one:

1) ∫sin(2x) dx:

We can use the identity ∫sin(ax) dx = -1/a * cos(ax) + C, where "a" is a constant.

In this case, a = 2, so the integral becomes:

∫sin(2x) dx = -1/2 * cos(2x) + C1

2) ∫(5tan^2x * csc^2x) dx:

Now, let's simplify this expression before integrating.

Using the identity tan^2x + 1 = sec^2x, we can rewrite tan^2x as sec^2x - 1:

∫(5tan^2x * csc^2x) dx = ∫(5(sec^2x - 1) * csc^2x) dx

Using the identity csc^2x = 1/sin^2x, we can further simplify:

∫(5*(sec^2x - 1) * csc^2x) dx = ∫(5(sec^2x - 1) * 1/sin^2x) dx

Expanding and rearranging terms, we get:

∫(5sec^2x / sin^2x - 5/sin^2x) dx

Now, using the identity sec^2x = 1 + tan^2x, we can substitute:

∫((5(1 + tan^2x)) / sin^2x - 5/sin^2x) dx

Simplifying further:

∫(5/sin^2x) dx = 5∫csc^2x dx = -5cotx + C2

∫(5tan^2x / sin^2x) dx = 5∫tan^2x csc^2x dx = 5(-cotx) + C3

Combining the results, we have:

∫(5tan^2x * csc^2x) dx = -5cotx - 5cotx + C = -10cotx + C4

Putting it all together, we have the final result:

∫(sin(2x) + 5tan^2x * csc^2x) dx = -1/2 * cos(2x) - 10cotx + C, where C = C1 + C2 + C3 + C4.