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ò 1/(x + Ö(1-x2))
By Colin Rowlands on Monday, May 06, 2002 - 11:41 am:

I've tried various substitutions which look promising but an algebraic solution to this integral still elludes me. Please help!

ò 1/(x + Ö(1-x2))

By David Loeffler on Monday, May 06, 2002 - 12:08 pm:

Try substituting x = sin(q). Then it is

I = ò cos q dq/(sin q + cos q).

Now this is a fairly well known, and moderately difficult, integral; it is one of the questions in Dr Siklos's book of practice STEP questions (IIRC). You can evaluate it in one of two ways: substitute t = tan q/2 and wade through lots of partial fractions; or by a cunning trick - define

I1 = ò cos q dq/(sin q + cos q)

I2 = ò sin q dq/(sin q + cos q)

Now I1 + I2 = ò dq = q + C; and I1 - I2 =

ò (cos q - sin q)/(sin q + cos q)

Now the numerator of the integrand is the derivative of the denominator; so I1 - I2 = log( sin q + cos q ) + D.

If you add these two equations together, we get

I = I1 = 1/2 q + 1/2 log( sin q + cos q ) + E

(E is some new arbitrary constant = 1/2(C+D).)

So that is your integral: 1/2 sin-1(x) + 1/2 log (x + Ö(1-x2) ) + E.

David