(ii) ∫ + C a dx (iii) Alternatively , integrals (i), (ii) and (iii) can also be found by making trigonometric substitution x = a sec θ in (i), x = a tan θ in (ii) and x = a sin θ in (iii) respectively. Example Find Solution Note that Put x + = y , so that dx = dy . Then y dy y y y y [using . .
(ii)] ( log x + + Example Find x Solution Note that Put x + = y so that dx = dy . Thus x y dy – y y y [using . . (iii)] sin EXERCISE .
Integrate the functions in Exercises to . . . x .
x . . Choose the correct answer in Exercises to . .
is equal to (A) (B) ( (C) ( x (D) . is equal to (A) ( ) 9log x (B) ( ) 9log x (C) ( ) 2log x (D) ) INTEGRALS . Definite Integral In the previous sections, we have studied about the indefinite integrals and discussed few methods of finding them including integrals of some special functions. In this section, we shall study what is called definite integral of a function.
The definite integral has a unique value. A definite integral is denoted by a f x dx , where a is called the lower limit of the integral and b is called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the interval [ a , b ], then its value is the difference between the values of F at the end points, i.e., F (b) – F (a) . .
Fundamental Theorem of Calculus . . Area function We have defined a f x dx as the area of the region bounded by the curve y = f ( x ), the ordinates x = a and x = b and x -axis. Let x be a given point in [ a , b ].
Then a f x dx represents the area of the light shaded region in Fig . [Here it is assumed that f ( x ) > for x ∈ [ a , b ], the assertion made below is equally true for other functions as well]. The area of this shaded region depends upon the value of x . In other words, the area of this shaded region is a function of x .
We denote this function of x by A( x ). We call the function A( x ) as Area function and is given by