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

I=òcosqdq/(sinq+cosq).

Now this is a fairly well know, 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=tanq/2 and wade through lots of partial fractions; or by a cunning trick - define

I₁=òcosqdq/(sinq+cosq)

I₂=òsinqdq/(sinq+cosq)

Now I₁+I₂=òdq = q+C; and I₁=I₂=ò(cosq-sinq) /(sinq+cosq)

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

If you add these two equations together, we get

I=I₁=1/2 q+1/2log(sinq+cosq)+E

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

So that is your integral:

1/2sin-1(x)+1/2log(x+

____ Ö1-x2

)+E

.

David