∫tan(5x)dx

Sure, here is the step by step solution for the integral of tan(5x) dx:

Step 1: We know that the integral of tan(x) dx is ln|sec(x)| + C. But we have tan(5x) instead of tan(x). So, we need to use the substitution method.

Step 2: Let u = 5x. Then, du/dx = 5 or dx = du/5.

Step 3: Substitute u = 5x and dx = du/5 in the integral. We get:

∫tan(u) * (1/5) du

Step 4: Now, this integral can be solved as:

(1/5) * ∫tan(u) du

Step 5: As mentioned before, the integral of tan(u) du is ln|sec(u)|. So, we get:

(1/5) * ln|sec(u)| + C

Step 6: Substitute u = 5x back in the above expression. We get:

(1/5) * ln|sec(5x)| + C

So, the integral of tan(5x) dx is (1/5) * ln|sec(5x)| + C.