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Advanced Math/integration of rational functions by partial fractions


I am doing this problem to review for a test and need a refresher on how to do it.

Make a substitution to express the integrand as a rational function and then evaluate the integral.

It is telling you to make a substitution. So first, the denominator of the fraction is a function of sinh(t), which should lead you to believe u=sinh(t) might be a good substitution.

The numerator is cosh(t), which makes this perfect. You get:

du = cosh(t)dt

and the integral becomes:

∫ 1 / (u^2 + u^4) du

Then you split up the fraction into:

1 / (u^2 + u^4) = A/u + B/u^2 + (Cu+D) / (u^2+1)

To solve for A,B,C, usually you clear out the denominator across the board:

1 = Au(u^2+1) + B(u^2+1) + (Cu+D)u^2

Plug in u=0 to get:

1 = B.

Plug in u=i (which is allowed, even if you normally worry only about real numbers) to get:

1 = -(Ci+D)

which means C = 0 and D = -1.

Finally, plug in u=1 to get:

1 = 2A + 2B + C + D

which (since you know B, C, and D already) gives you A = 0.

So you have:

1 / (u^2 + u^4) = 1/u^2 - 1/(u^2+1)

From there, just integrate this to get:

-1/u - arctan(u)

and finally plug in for u to get:

-1/sinh(t) - arctan(sinh(t))

which can be rewritten:

- csch(t) - arctan(sinh(t))

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Clyde Oliver


I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.


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