What is the correct solution for this integral?

[alba_ei]

Homework Statement

$$\int \frac{ae^\theta+b}{ae^\theta-b} \, d\theta$$

The Attempt at a Solution

i took $$u = ae^\theta-b$$ so $$e^\theta = \frac{u + b}{a}$$ then i substituded back into the integral and iget this

$$\int \frac{u + b + b}{u} \, du$$

$$\int du +\int \frac{2b}{u} \, du$$

$$= u \du + 2b \ln u +C$$

$$= u + 2b \ln u +C$$

$$= ae^\theta-b + 2b\ln (ae^\theta-b)$$

but the answer of the book is
$$\int \frac{ae^\theta+b}{ae^\theta-b} \, d\theta = 2\ln (ae^\theta-b) - \theta + C$$
what did i do wrong?

 

[Galileo]

You didn't subsitute properly. You have to change dtheta too.

 

[ziad1985]

write d0 as what it should equal to du
for example if u=x^2
du=2*xdx