| ∫ | dx |
| (x − p)n √(ax2 + 2bx + c) |
can be deduced by differentiating (n − 1) times with respect to p.
(iii.)
| ∫ | (Hx + K) dx |
| (αx2 + 2βx + γ) √(ax2 + 2bx + c) |
can be reduced by the substitution y2 = (ax2 + 2bx + c)/(αx2 + 2βx + γ) to the form
| A ∫ | dy | + B ∫ | dy |
| √(λ1 − y2) | √(y2 − λ2) |
where A and B are constants, and λ1 and λ2 are the two values of λ for which (a − λα)x2 + 2(b − λβ)x + c − λγ is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888).
(iv.) ƒxm (axn + b)p dx, in which m, n, p are rational, can be reduced, by putting axn = bt, to depend upon ƒtq (1 + t)p dt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1 + t = us. If p + q is an integer and p = r/s we put 1 + t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).
(v.)
| ∫ | dx | = log tan | x | , |
| sin x | 2 |