2 z a a f x dx 0 3 z b a f x dx z c b f x dx z c a f

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(2) Z a a f ( x ) dx = 0. (3) Z b a f ( x ) dx + Z c b f ( x ) dx = Z c a f ( x ) dx . (4) Z b a kf ( x ) dx = k Z b a f ( x ) dx . (5) Z b a { f ( x ) + φ ( x ) } dx = Z b a f ( x ) dx + Z b a φ ( x ) dx . The reader will find it an instructive exercise to write out formal proofs of these properties, in each case giving a proof starting from ( α ) the definition by means of the integral function and ( β ) the direct definition. The following theorems are also important. (6) If f ( x ) = 0 when a 5 x 5 b , then Z b a f ( x ) dx = 0 . We have only to observe that the sum s of § 156 cannot be negative. It will be shown later ( Misc. Ex. 41) that the value of the integral cannot be zero unless * All functions mentioned in these equations are of course continuous, as the definite integral has been defined for continuous functions only.
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[VII : 160] ADDITIONAL THEOREMS IN THE CALCULUS 359 f ( x ) is always equal to zero: this may also be deduced from the second corollary of § 121 . (7) If H 5 f ( x ) 5 K when a 5 x 5 b , then H ( b - a ) 5 Z b a f ( x ) dx 5 K ( b - a ) . This follows at once if we apply (6) to f ( x ) - H and K - f ( x ). (8) Z b a f ( x ) dx = ( b - a ) f ( ξ ) , where ξ lies between a and b . This follows from (7). For we can take H to be the least and K the greatest value of f ( x ) in [ a, b ]. Then the integral is equal to η ( b - a ), where η lies between H and K . But, since f ( x ) is continuous, there must be a value of ξ for which f ( ξ ) = η ( § 100 ). If F ( x ) is the integral function, we can write the result of (8) in the form F ( b ) - F ( a ) = ( b - a ) F 0 ( ξ ) , so that (8) appears now to be only another way of stating the Mean Value Theorem of § 125 . We may call (8) the First Mean Value Theorem for Integrals . (9) The Generalised Mean Value Theorem for integrals. If φ ( x ) is positive, and H and K are defined as in (7) , then H Z b a φ ( x ) dx 5 Z b a f ( x ) φ ( x ) dx 5 K Z b a φ ( x ) dx ; and Z b a f ( x ) φ ( x ) dx = f ( ξ ) Z b a φ ( x ) dx, where ξ is defined as in (8) .
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[VII : 160] ADDITIONAL THEOREMS IN THE CALCULUS 360 This follows at once by applying Theorem (6) to the integrals Z b a { f ( x ) - H } φ ( x ) dx, Z b a { K - f ( x ) } φ ( x ) dx. The reader should formulate for himself the corresponding result which holds when φ ( x ) is always negative. (10) The Fundamental Theorem of the Integral Calculus. The function F ( x ) = Z x a f ( t ) dt has a derivative equal to f ( x ) . This has been proved already in § 145 , but it is convenient to restate the result here as a formal theorem. It follows as a corollary, as was pointed out in § 157 , that F ( x ) is a continuous function of x . Examples LXV. 1. Show, by means of the direct definition of the definite integral, and equations (1)–(5) above, that (i) Z a - a φ ( x 2 ) dx = 2 Z a 0 φ ( x 2 ) dx , Z a - a ( x 2 ) dx = 0; (ii) Z 1 2 π 0 φ (cos x ) dx = Z 1 2 π 0 φ (sin x ) dx = 1 2 Z π 0 φ (sin x ) dx ; (iii) Z 0 φ (cos 2 x ) dx = m Z π 0 φ (cos 2 x ) dx , m being an integer. [The truth of these equations will appear geometrically intuitive, if the graphs of the functions under the sign of integration are sketched.] 2. Prove that Z π 0 sin nx sin x dx is equal to π or to 0 according as n is odd or or even. [Use the formula (sin nx ) / (sin x ) = 2 cos { ( n - 1) x } +2 cos { ( n - 3) x } + . . . , the last term being 1 or 2 cos x .] 3. Prove that Z π 0 sin nx cot x dx is equal to 0 or to π according as n is odd or even.
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