Зображення сторінки
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cos x = (cos x)2 - (sin x), and 1 = (cos x)2 + (sin x)2,

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2 √ a + b + (a - b) **:

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and may therefore be integrated by (c). The result is

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dx

2

α b

a + b cos x

tan

-1

(a2 — b2)}

tan

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Integration by parts often decomposes a function into an integrated part and one easily integrated. The general for

mula is

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2 a 8

}

(10) fdx tan-1 x = æ tan ̄1 æ

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tan-1= (-tana) tan-1a-log (1 + a2).

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On comparing these expressions with the formulæ in Ex. (10) of Chap. 11. Sec. 1, of the Diff. Calc. it will be seen that they may be deduced from the latter by making r=-1.

U

If

V

Rational Fractions.

be a rational fraction, in which the numerator is of lower dimensions than the denominator, it may always be decomposed into a sum of simpler fractions differing according to the form of V.

V may consist of factors of the forms.

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I. To every factor of the form a corresponds a

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