Integration of an equation by hand

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The discussion revolves around the integration of a velocity equation, with the user initially struggling to find the correct solution. They attempted to solve for a constant under specific initial conditions but were unsure of their result. A hint was provided to use integration by parts for the integral of tan(ax+b). Ultimately, the user confirmed that they successfully solved the problem. The conversation highlights the importance of integration techniques in solving complex equations.
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Homework Statement



I have some trouble integrating this velocity equation http://imgur.com/Ope9R


The Attempt at a Solution


i have tried myself and have come to this equation after solving for c for the condition t,s=0,s_0 however it is fine if the condition is t,s=0,0
http://imgur.com/wkuWW
This does not seem to be correct though.
 
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What's the value of ∫tan(ax+b)dx ?

Hint: Solve it using integration by parts.
 
Last edited:
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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