Is this the correct solution to the given antiderivative problem?

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Homework Statement


f''(x)=x^-2, x>0, f(1)= 0 f(2)=0 Find f(x)


Homework Equations





The Attempt at a Solution



f'(x)= -x^-1 +C , f(x)= -lnx + Cx + D ... after working out i got:

f(x)= -lnx + (ln2)x - ln2 ... does this seem correct
 
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Well, there's an easy way to check whether something is a solution to an equation: substitute it in and see if it satisfies the differential equation!
 
Looks right to me.
 

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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