Possible Integration by Parts Exam Practice Problems

Goldenwind
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Homework Statement


Got an exam later today on this, just looking for some practice.
I know this is kinda reverse of what the forum is intended for, but I'd like to ask for people to post some problems that I can solve via Integration by Parts.

Just like 5 or so.

Homework Equations


Integral(u dv) = uv - Integral(v du)
 
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\int\frac{r^3}{\sqrt{4+r^2}}dx

\int\ln \sqrt{1+x^2}dx

\int\sec^{3}xdx

\int\sec^{5}xdx

\int x\tan^{-1}xdx
 
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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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