Integration of (tanx-tana)/(tanx tana)

To integrate (tanx-tana)/(tanx tana), we need to use substitution method. Here are the steps:

1. Substitute u = tanx. This means that du/dx = sec^2x, or dx = du/sec^2x.
2. Substitute tana = 1/u. This means that da/du = -1/u^2, or du = -da/u^2.
3. Substitute these expressions into the original integral to get:

∫[(u - 1/u)/(u*1/u)] * (-da/u^2) / sec^2x dx

4. Simplify the expression to get:

-∫(u^2 - 1)/a * sec^2x dx

5. Use the identity sec^2x = 1 + tan^2x to get:

-∫(u^2 - 1)/(u^2 + 1) du

6. Now we can use partial fraction decomposition to split the integrand into two terms:

(u^2 - 1)/(u^2 + 1) = A(u-1)/(u^2+1) + B(u+1)/(u^2+1)

7. Solving for A and B, we get:

A = -1/2 and B = 1/2

8. Substituting these values back into the integral, we get:

-∫(1/2)*(u-1)/(u^2+1) du + ∫(1/2)*(u+1)/(u^2+1) du

9. Using u-substitution again, we get:

-(1/2)ln|u^2+1| + (1/2)ln|u^2+1| + C

10. Substituting back u = tanx and tana = 1/u, we get:

-(1/2)ln|tan^2x+1| + (1/2)ln|1+tan^2x| + C

Conclusion

Therefore, the final solution to the integration of (tanx-tana)/(tanx tana) is -(1/2)ln|tan^2x+1| + (1/2)ln|1+tan^2x| + C.