Integration by Parts and Substitution: Solving Complex Integrals

robertjford80
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Homework Statement



Screenshot2012-05-19at80808PM.png



Homework Equations



uv - integral of vdu

The Attempt at a Solution



They don't seem to be using the integration by parts formula here. I don't understand why why they don't have a value for what z equals. dz = eu. well, what does z equal. I would think it would be the same thing. Next I don't anywhere where they're using the uv - integral of vdu formula. very bizarre.
 
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You're quite right for what z is. As for integration by parts, they are using it - I recommend looking up tabular integration.
 
actually, i relooked at the question and they said they want me to use substitution in combination with integration by parts, so I'm going to have to look at some of the other examples in the book and see if they have a worked example for this type of problem.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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