36 DIFFERENTIAL AND INTEGRAL CALCULUS. 



1 1 X 1.3 1.3.5 



therefore - - = - + - * + x +... ; 



therefore 



1.3.5 1 



2 2 2.4 2 2.4.6 3 



d0 coB0J0\ * 



tan - 

 Bin0 



/* 



I 



J 



= 2 log (sec-) ^ = 2 log V(2) =log2. 

 \ ^/ 



QUESTIONS ON INTEGRAL CALCULUS. I. 



1. If V(x) = <t>(x), and = ^ (x), prove that 



7=^(0:)+ (7 where (7 may have any value independent 

 of x. Explain what is meant by an indefinite, a correct </, 

 and a definite integral. 



2. If i/r' (a?) = $ (a;), and A=-(&-a), prove that the 



i'wzV of the sum of the n elements 



(a) 4 * (a + A) + <f> (a + 2h) +...4 { a + (n- 

 when h = 0, is ty (b) ^r (a). Hence prove that the sum 



1 1 _J. _ 1 



n + V(w a -l 2 ) + VK - 2 a ) + * ' + V{w 2 - (n - I 2 )} 



tends to the limit ^TT when n = x . Find the area of the 

 curve y* = 4aic included between the ordinates x = 2a, a=8, 

 and the curve. 



3. Write down the integrals of # m , - , a*, wix , sin a?, 



CB 



C03a; ' secV ' ^ 5 ' <7T^ ' ; also deduce thc 



/dx ( x m ^ 



from the equation lx m dx = - - + C. 

 x J m4 1 



