By E. A. Maxwell

This is often the second one of a chain of 4 volumes overlaying all phases of improvement of the Calculus, from the final years in class to measure ordinary. The books are written for college students of technological know-how and engineering in addition to for professional mathematicians, and are designed to bridge the space among the works utilized in colleges and extra complex reports. with their emphasis on rigour. This remedy of algebraic and trigonometric features is the following constructed to hide logarithmic, exponential and hyperbolic services and the growth of these kind of services as strength sequence. there's a bankruptcy on curves and the belief of complicated numbers is brought for the 1st time. within the ultimate chapters, the writer starts off a scientific therapy of equipment of integrating capabilities, introducing ideas into what usually turns out fairly a haphazard approach. This quantity, just like the others, is definitely endowed with examples.

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**Sample text**

By giving k other values, we obtain various forms for the remainder. In particular, when k = 1, we have = "~ (6-q)(6-g)n-i J (n-1)! ( } * This may be expressed alternatively by writing f, which lies between a, b, as a + 6(b—a), where 6 lies between 0 and 1. Then This is called Cauchy's form of the remainder. Though less 'in sequence' than Lagrange's form, it enables us to deal with some series for which Lagrange's form does not work. 48 T A Y L O R ' S S E R I E S AND A L L I E D R E S U L T S ALTERNATIVE TREATMENT.

If we put k = n (as we might have done from the start, of course, had we so desired) we have (6-q)» TAYLOR'S THEOREM 47 and the result enunciated at the head of the paragraph follows at once. This gives us the most usual form for Taylor's theorem, and we have adopted it in the formal statement; but there are advantages in keeping k more general, as we see below. The theorem is therefore established. The expression is called the remainder after n terms. It is, in the first instance, what it says, namely a remainder, the difference between Kb) lh — n\n-l (h — n\% and / The theorem just proved enables us, however, to express this remainder in the suggestive form (with k = n) >(0 by choosing £ suitably.

43. Integrate with respect to x: 44. Integrate with respect to x: [In sii Jo sin 3# cos 2x dx. Evaluate 45. Explain the method of integration by parts, and employ it to integrate ^(1 — x2) with respect to x. Integrate with respect to x sin*"1^ 171 1\> ](l-X2) 1 Ti 2T#> (1X2)$ . ex(cosx-smx). 46. Integrate with respect to x: . 5 x2+l 47. Integrate with respect to #: 1 sin2 # + 2 cos2 #' 1 a; 3 +l # By integration by parts, or otherwise, integrate sin-1 x. REVISION EXAMPLES III 35 48. Integrate with respect to x: 1-x' Prove that (1 — cos2 x)2 sin x, -, xex.