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## Homework Statement

[tex]\int \frac{ae^\theta+b}{ae^\theta-b} \, d\theta[/tex]

## The Attempt at a Solution

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

[tex]\int \frac{u + b + b}{u} \, du[/tex]

[tex]\int du +\int \frac{2b}{u} \, du[/tex]

[tex]= u \du + 2b \ln u +C[/tex]

[tex]= u + 2b \ln u +C[/tex]

[tex]= ae^\theta-b + 2b\ln (ae^\theta-b) [/tex]

but the answer of the book is

[tex]\int \frac{ae^\theta+b}{ae^\theta-b} \, d\theta = 2\ln (ae^\theta-b) - \theta + C [/tex]

what did i do wrong?

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