## A treatise on the integral calculus and its applications1857 |

### From inside the book

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**hence**as the two members are identical we can divide by this factor . Let ( x ) indicate the quotient obtained on the left ; then - ¤1 ( x ) = ( L ̧_ ̧x + M‚_1 ) „ ( x ) + ( L‚_2x + Mr_2 ) ( x2 — px + q ) † ( x ) + ...... + ( x2 - px + ... Page 33

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**Hence**corresponding to the root 1 we have the partial fraction 1 - n ( x − 1 ) ' the partial fraction of roots and corresponding to the root - 1 we have ( -1 ) m n ( x + 1 ) And corresponding to the pair cos re ± √ ( -1 ) sin re ... Page 34

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**Hence**xm - 1dx 1 xn 1 - n 1log ( x log ( x - 1 ) + ( -1 ) m log ( x + 1 ) n 1 n + -cosmre log ( x2 −2x cosre + 1 ) . 2Σsinmr @ tan -1 n x - cos re sin re 27. Required the integral of xm - 1 n being supposed odd . 2 x " - 1 The real ... Page 35

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**Hence**2 cos mr0 ( — cos r0 ) — sin mr0 sin re n - 2 ( x — cos r0 ) 2 + sin2r0 dx n == n Σ cos mre log ( x2 - - 2x cos re + 1 ) -1 х cos re sin re " + Σ sin mrš tan TM 1 where indicates a sum to be formed by giving to r all the odd ... Page 42

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**Hence**x * - - x ) -1dx 0 = r x2 + " = 1 fœ ̃ ̄ * ( 1 − x ) " dx . n — 1 - x2 ( 1 - x ) " dx . n 0 Thus if r be an integer we may reduce the integral to ( 1 - x ) dx , that is - x ) n - 1 dx = ― 1 n + r 1 -**hence**( r− 1 ) ( r− 2 ) ...### Other editions - View all

### Common terms and phrases

Application c² sin² Cambridge circle cloth co-ordinates constant cos² cos³ Crown 8vo curve cycloid definite integral denote differential coefficient double integral dx dx dx dy dz dy dx dz dx element ellipse equal Eulerian integral example expression Fcap Fellow of St Find the area find the volume formula function Hence indefinitely Integral Calculus integrate with respect intrinsic equation John's College latus rectum length limits M.A. Fellow MACMILLAN & CO.'S numerous obtain ordinate parabola partial fractions perpendicular plane positive preceding article radius vector result revolution revolve round round the axis shew sin² solid suppose surface tangent tion tractory transformed integral Trinity College unity vanishes variables vertex x₁ Y₁ πα аф

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