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This integral came up while trying to find the potential of a uniformly charged rectangle.

[tex]\int \log(\sqrt{a^2+x^2} + b) dx[/tex]

Integrator gives a pretty long expression involving inverse tangents so I'm not sure where to begin at all. I tried integrating by parts once, taking u to be the whole expression, but it just makes it messier. I also tried the trig subtitution:

[tex]x = a \tan(\theta)[/tex]

[tex]\int a \log(a \sec(\theta) + b) \sec^2(\theta) d \theta[/tex]

But that's not any easier to integrate.

[tex]\int \log(\sqrt{a^2+x^2} + b) dx[/tex]

Integrator gives a pretty long expression involving inverse tangents so I'm not sure where to begin at all. I tried integrating by parts once, taking u to be the whole expression, but it just makes it messier. I also tried the trig subtitution:

[tex]x = a \tan(\theta)[/tex]

[tex]\int a \log(a \sec(\theta) + b) \sec^2(\theta) d \theta[/tex]

But that's not any easier to integrate.

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