How to carry units in integration

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Homework Statement


I am trying to integrate from 0 to 10 ft -4x^2/2+4x. I am solving for deflection. The forces are in kips. Somehow the book ends up with k-ft^3. I don't understand how I would carry units in an integration. Any help please?


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The Attempt at a Solution

 
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Normally integrating gives an area which would give the product of the units on the axes. I'd be inclined to think you'd get kip-ft. Are you sure the units are correct?
 

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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