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(17) Salatbest) - sa log (a to op) (18) Sem ca tots somit na bang ( t). (19) Socatibey - saku+bah war log (* bat). (20) Septemba 100 (1 =* * 9) + 3 tan-" (**). (21) si log (1 +)'+ £ tan-'c. (22) Sid = 105 (1 + ) – 4 tan-' «. (23) Sale 410.- ha log (* * ). (es) s *(***

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If an integral be a function of several fractional powers of w, it may be rationalized by assuming x = z", r being equal to the product of the denominators of the indices.

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When the integral involves also fractional powers of binomials, such as a + bx, it can be rationalized by assuming a + bx = x", go being the product of the denominators of the indices. If the binomials be of the form a +ba", they may be reduced to the preceding form by assuming x* = y.

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If the function to be integrated involve (a + bx + cx')} it may be reduced to the preceding forms, as

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11 – X – 2 (1 + x + x?)) (1 + x) (1 + x + x“).

1 + x dx .

ir 1 + 32 ] (31) J (1 + x) (1 + 2 – 12°)= tan- ' 2 (1 + 2 – 2)] (35) _ dx (35) / (1 + x) (1 – 2 – x;)? = log

(3 + x – 2(1 – X – 0??))

ita ) Various functions can be rationalized by assumptions for which no general rule can be given : familiarity with the transformations to which different substitutions lead is the best way of acquiring a knowledge of the most convenient assumption in particular cases.

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