Definite Integral of Both Sides

[SherwinS]

Homework Statement

"Integrate both sides of these equations from some starting point to some ending point (such as t0 to t): r dt = d[A], C dt = dH and dw = (nRT/V) dV

Homework Equations

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The Attempt at a Solution

I'm have trouble understanding the concept of taking the definite integral of both sides using the same starting points for both sides. If someone could guide me on how to do one of these, I'll be able to finally do the rest. Thanks!

As a side note, can someone check my integration of these? :)
dy = 3 sin 2x dx ---> -3/2 cos 2x + C
df = -2 cos (x/2) dx ---> 4 sin (x/2) + C
dP = A sin^2 (kx) dx ---> A/2 (x - 1/2ksin (2kx)) + C

 

[jedishrfu]

If you differentiate your three solutions then you could check them yourself.

The first two look okay. The third looks ambiguous. Is the sin in the numerator or denominator?

Maybe you can show your attempt before someone shows you how.

 

[SherwinS]

Sorry, 1/2k is the coefficient, sin is in the numerator. Yes I've differentiated them already and come up with the right result, but I always feel it's better to have another pair of eyes glaze over them. :)