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If a1, a2, ag, etc., are the roots of the equation

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MISCELLANEOUS ALGEBRAIC EXPRESSIONS.

33

where m<n, may be expressed as the sum of a number of

partial fractions of the form

A

(x − a) √ a + bx + cx2

, and these

can be integrated by the aid of equations given above. Thus,

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34

MISCELLANEOUS ALGEBRAIC EXPRESSIONS.

232. Sƒ (x, Ŵc+Wa+bæ) dx

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where y' (a' + b'x) = a + bx and s is the least common multi

ple of n, q, etc.

234. Sƒ(x, √a+bx+ x2) dæ

́2 √ a · z − b ̧ z2 √ a − bz +√a) (22 √a − bz +√a) dz

=2Ss(2

1 22

-

1 22

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(b − 2 u)2

V. TRANSCENDENTAL FUNCTIONS.

sin x · ƒ (cos x) dx = —

236. Ssi

-Sf

f(cos x)d cos x.

=Sf

Ssin

237. Sc cos x · ƒ (sin x) dx = ƒ ƒ (sin æ) d sin æ.

238.

.

sin æ⋅f (sin x, cos x) dx = −

where = cos x.

Sƒ(√1 — 22, ñ)dz,

239. Sco
cos x ⋅ƒ (sin x, cos x) dæ = ƒ ƒ (≈, √1 − 22) dã,

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where z = sin x.

240. Sƒ (sin x, cos x)
x, cos x) dx = ƒ ƒ (≈, √1 − 2o)

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π

π

242. Sƒ (tan x) dx = — -Setn (1-x)d (1-x)

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fctn

2

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sin x f(sin x) dx

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V1-2 sin2x

where sin2x.

245. S

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'cos x ⋅f (cos2x) dx

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√1 - k2 sin2x

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247. S ƒ (tan x) dx = Sf (2) dz, where z = tan x.

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250. ƒ ƒ (2) · sin ̄1æ · dæ = sin−1 x · 4 (x) − f (x) da

х

f(sin x, cos x) dx

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dx V1 - x2

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dx,

252. fr()

where 4(x) = ƒ ƒ (x)dx.

251. Sf(x)·cos
· cos ̄1 x dx = cos ̄1x · p (x) + f (x) dx,

x ·

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