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

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**Hence**we may put where P1 , P2 , when h is indefinitely diminished is ↓ ( x1 ) — ( a ) = h1 { $ ( a ) + p1 } ...**hence**Zho ultimately vanishes , and we obtain this result - the limit of Σ ( x ) Ax when each of the quantities of which Ax ... Page 5

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**hence**if ( x ) be a function having ( x ) for its differential coeffi- cient , then ( x ) + C , where C is any quantity independent of X , is the only form that can have the same differential coefficient .**Hence**, hereafter , when we ... Page 10

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**Hence**Ex . ( 3 ) . dx √√ ( x2 + a2 ) = z − X , dx dz a2 = z2 — 2zx , 2 - 2 1 х - dx dz √ √ ( x2 + a2 ) = √ √ ( x2 + a2 ) dz dx - 7 . √√ ( x2 — a2 ) √ ( x2 = = log { x + √ ( x2 + a2 ) } . As in Ex . ( 2 ) , we may shew that ... Page 14

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**Hence**, if a be greater than b , the integral is x 2 -1 tan √ ( a2 — b2 ) -- z √ ( a - b ) √ ( a + b ) 2 or Ja and if a be less than b , 1 √ ( a2 — b2 ) tan 2.√ ( a - b ) tan -1 √ ( a + b ) z √ ( b − a ) + √ ( b + a ) ... Page 15

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**Hence**for ( a + bx " ) } " dx = fx m 2dx dt ( a + bx " ) de de dt - 2 Jors ( 12 - a ) nb ግን n dt . m -1 If be a positive integer we can expand ( ťa — a ) " ̄1 in n a finite series of powers of t , and each term of the product of this ...### 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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