Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

The document Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Definite Integral As Limit Of A Sum

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Remark : The symbol  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev was introduced by Leibnitz and is called integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. The symbol dx has no official meaning by itself;  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  is all one symbol. The procedure of calculating an integral is called integration.


Ex.26 Evaluate Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  using limit of sum.

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

This integral can't be interpreated as an area because f takes on both positive and negative values. But is can be interpreated as the difference of areas A1 – A2, where A1 and Aare shown in Figure

Estimate Of Definite Integration & General Ineθuality

STATEMENT : If f is continuous on the interval [a, b], there is atleast one number c between a and b such that  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Proof : Suppose M and m are the largest and smallest values of f, respectively, on [a, b]. This means m ≤ f(x) ≤ M when  a ≤ x ≤ b  

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Because f is continuous on the closed interval [a, b] and because the number I = Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev lies between m and M, the intermediate value theorem syas there exists a number c between a and b for which f(c) = I ; that is, Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

The mean value theorem for integrals does not specify how to determine c. It simply guarantees the existence of atleast one number c in the interval.
Since f(x) = 1 + x2 is continuous on the interval [–1, 2], the Mean Value Theorem for Integrals says there is a number c in [–1, 2] such thatDefinite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

In this particular case we can find c explicitly. From previous Example we know that fave = 2, so the value of c satisfies f(c) = Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev = 2

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Thus, in this case there happen to be two numbers c = ± 1 in the interval [–1, 2] that work in the mean value theorem for Integrals.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Walli's Formula & Reduction Formula

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Ex.27 Prove that,  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev Hence or otherwise evaluate Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

 Consider sin nθ + sin (n - 2) θ = 2 sin (n - 1) θ cos θ ⇒ sin nθ sec θ = 2 sin (n-1) θ - sin (n - 2) θ sec θ

Hence  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev,Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Ex.28 Prove that  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

L.H.S. =  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev (From Walli's formula)


Ex.29 If un =  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev   then show that u1, u2, u3 ,...... constitute an arithmetic progression. Hence or otherwise find the value of un.

Sol.

un + 1 – 2un + un – 1 = (un + 1 – un) – (un – un – 1)

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

∴ un – 1 + un + 1 = 2ui.e., un – 1, un, un + 1 form an A.P.

⇒ u1, u2, u3,.........constitute an A.P.
Ex.30 Evaluate  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Integrating by parts taking unity as the second function, we have

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Ex.31 Show that  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Hence or otherwise evaluate Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

adding (1) and (2) then 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Put a tan x = t ⇒ a sec2x dx = dt when x = 0 ⇒ t = 0 ;  x = p/2 ⇒ t = ∝

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Differentiating both side w.r.t. 'a', we get   Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

again differentiating both sides w.r.t. 'a' we get  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Put a = √5 on both sides, we get

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev
Ex.32 Let f be an injective functions such that f(x) f(y) + 2 = f(x) + f(y) + f(xy) for all non negative real x and y with f(0) = 1 and f '(1) = 2 find f(x) and show that Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRevf(x) dx – x (f(x) + 2) is a constant.


Sol. We have f(x) f(y) + 2 = f(x) + f(y) + f(xy)                          ......(1)
Putting x = 1 and y = 1 then f(1) f(1) + 2 = 3f(1)
we get f(1) = 1, 2 & f(1) Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev 1           ( Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev f(0) = 1 & function is injective) then f(1) = 2

Replacing y by 1/x in (1) then f(x) f(1/x) + 2 = f(x) + f(1/x) + f(1) ⇒  f(x) f(1/x) = f(x) + f(1/x) [ f(1) = 2)

Hence f(x) is of the type f(x) = 1 ± xn ⇒ f(1) = 1 ± 1 = 2 (given)

∴ f(x) = 1 + xn  and f '(x) = nxn – 1 ⇒ f '(1) = n = 2 ∴ f(x) = 1 + x2

∴  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Ex.33 Evaluate  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Sol.

Let  I = Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev Make |sin x| – |cos x| = 0 ∴  |tan x| = 1

∴ tan x = ± 1  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  and both these values lie in the interval [0, π].

We find for 0 < x < π/4, |sin x| – |cos x| < 0 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  |sin x| – |cos x| > 0 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Ex.34 Evaluate  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev, (where [ * ] is the greatest integer function)

Sol. Let I = Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Let f(x) = x2 + x + 1  ⇒  f '(x) = 2x - 1 for x > 1/2, f '(x) > 0 and x < 12, f '(x) , 0

Values of f(x) at x = 1/2 and 2 are 3/4 and 3 integers between them an 1, 2 then x2 - x + 1 = 1, 2

we get x = 1,  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev and values of f(x) at x = 0 and 1/2 are 1 and 3/4 no integer between them

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Alternative Method :  It is clear from the figure 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Ex.35 If  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Integrating by parts taking x2 as 1 st function, we we get  =  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev [By Prop.] 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Adding (1) and (2) we get  

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

 

Ex.36 Prove that  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev Intergrating by parts taking x as a first function, we have 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Ex.37 Evaluate  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Let g(t) = |t – 1| – |t| + |t + 1| =  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Ex.38 For all positive integer k, prove that Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Hence prove that Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Sol. We have 2 sinx [cos x + cos 3x + ...+ cos (2k–1)x ]
= 2 sinx cosx + 2 sinx cos 3x + ....+ 2 sinx cos(2k – 1)x
= sin 2x + sin 4x – sin 2x + sin 6x – sin 4x + ....... + sin 2kx – sin (2k – 2) x
= sin 2kx
⇒ 2[cos x + cos 3x + ........ + cos (2k – 1) x ] = sin2kx/sinx

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev [2 cos2x + 2 cos 3x cos x + ....... + 2 cos (2k – 1 ) x cosx ] dx

 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev [cosx + cos 3x + ....... + cos (2k – 1 ) x ]cosx dx

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev
 

Ex.39 Let f(x) is periodic function such that Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev Find the function f(x) if (1) = 1 .

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

from (2) and (3)  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Differentiating bot sides w.r.t.x, we get

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

then f(x) = 1/x3 But given f(x) is a periodic function Hence f(x) = 1


Ex.40 AssumeDefinite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev   then prove that
Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev


Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev 

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev
Ex.41 Use induction to prove that , Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Sol.

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Now sinkx = sin[(k + 1) x – x ] = sin(k + 1) x cos – cos(k + 1)x sin x 

Hence sin (k + 1 ) x cosx = sinkx + cos(k + 1) x sin x

Subistuting P(k + 1) =  Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Definite Integral As Limit Of A Sum And Estimate Of Definite Integrals JEE Notes | EduRev

Now I. B. P. to get the result

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