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

### From inside the book

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Page 7

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**formula**is called " integration by parts . " For example , consider fa For example , consider x cos ax dx . Since 1 d . sin ax cos ax = we may write the proposed expression in the form x a dx INTEGRATION BY PARTS . 7. Page 8

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**formula**, is equal to x sin ax sin ax dx dx • α x sin ax a dx a - / sin ax dx a x sin ax cos ax = + a a2 Again , J x2 d . sin ax cos ax dx - dx a dx x2 sin ax 2x sin ax dx a a x2 sin ax 2x d . cos ax + dx a a2 dx x2 sin ax 2x cos ax 2 ... Page 38

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**formula**of reduction ; by means of it we make the integral of 1X depend on that of x + n - 1 X - 1 . In the same way the latter integral can be made to depend on that of + 2n - 1 XP - 2 ; and thus , if p be an integer we may proceed ... Page 40

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**formula**or may obtain the required result inde- pendently . Thus , suppose we require xm - 1 dx - Jd ' d √ ( c2 — x2 ) dx - = xm- dx — m - 2 - = = − √ ( co = − - x2 ) x12 + ( m − 2 ) m - 2 — xo ) x TM -2 + − x2 ) x12 + ( m − 2 ) ... Page 41

Isaac Todhunter. This

Isaac Todhunter. This

**formula**will serve to reduce the form ( Ax + B ) dx ( x2 - · 2ax + a2 + B2 ) " " which occurs in Art ...**formula**becomes applicable . 33. These formulę of reduction are most useful when the integral has to be taken ...### 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

### Popular passages

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