10. If F(x) be an algebraical polynomial of less than n dimensions nearly equal to unity, n being a positive quantity. 13. Evaluate [" (a cos 0 + b sin @) log (a cos2 0 + b sin2 0) de. 0 supposing a greater than b. where a and b are positive, but a and B positive or negative; and shew that it is wholly real when α B b 16. Prove that ୮ П cot1 (1−x + x2) dx = log 2. dx 17. Prove that log (+) = log 2. 0 1 + x2 「, b2 tan1 m√2 cot-1 π √(1+m2) m 21. Shew that (ex — e ̄x2) dx = (b − a) √√π. 0 (Solutions of Senate-House Problems, by O'Brien and Ellis, p. 44.) in which the sign of the square root is always taken so as to make the quantity in the denominator positive. √(1+a+ y de dy = (-1), x2 the integral being extended over all the values of x and y which make x2 + y2 not greater than unity. the number of variables being n, and the integration being extended over all values which make x2 + y2+ z2+...... not greater than unity. |