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

### From inside the book

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**expression**all values from 1 to n differs from the required area of the curve by the sum of all the portions similar to the triangle PQp , and as this last sum is obviously less than the greatest of the figures of which PMNQ is one , we ... Page 7

... of this 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

... of this 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

Isaac Todhunter. we may write the proposed

Isaac Todhunter. we may write the proposed

**expression**in the form x d . sin ax dx , a dx and this , by the 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 ... Page 9

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**expressions**which we assert to be equal can only differ by a constant ; for if we differentiate both we arrive at the same result , namely 4 , ( x ) + ( x ) . Ex . ( 1 ) . √√ ( a2 — x2 ) dx . - - x2 ) [ √ ( a2 — x ° ) dx = x √ ( a2 ... Page 16

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**expressions**are immediately integrable . Ex . ( 1 ) . fœ2 ( a + x ) 3dę . m Here - n 3 : assume a + x = t ; the integral becomes - 2 f ( t − a ) 3t dt or 2 f ( t − 2at + a ° f ) dt , + 5 3 ) ; which gives ( ť 2 { - 2at a2t3 ) thus ...### 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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