OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

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OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-16(a), Exe-16(b), Exe-16(c), Exe-16(d), Self Revision and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

 Definite Integral ISC Class-12 Maths OP Malhotra Solutions Ch-16

Class: 12th
Subject: Mathematics
Chapter  : Ch-16 Definite Integral of Section -A
Board ISC
Topics Exe-16(a) , Exe-16(b), Exe-16(c), Exe-16(d) ,Self Revision ,Chapter Test
Writer  OP Malhotra, SK Gupta, Anubhuti Gangal
Publications S.Chand Publications 2020-21

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Definite Integral Definition :-

If an integral has upper and lower limits, it is called a Definite Integral. There are many definite integral formulas and properties. Definite Integral is the difference between the values of the integral at the specified upper and lower limit of the independent variable. It is represented as;

∫ab f(x) dx

Define definite integral :-

The definite integral is defined as an integral with two specified limits called the upper and the lower limit. The definite integral of a function generally represents the area under the curve from the lower bound value to the higher bound value.

Different properties of definite integral :-

Some of the important properties of definite integrals are:
Adding Function Property
Adding intervals property
Interval of zero-length property
Reversing the interval property
Area above – area below property

Difference between the definite integral and indefinite integral :-

The definite integral f(x) is a number which defines the area under the curves within the specified limits. It has an upper limit and lower limit and it gives a definite answer. Whereas the indefinite integral f(x) is a function and it has no upper and lower limits. It gives a solution to the question “what function produces f(x) when it is differentiated?”.

The definite integral of a function :-

The definite integral of a function on the interval [a, b] is defined as the difference of antiderivative of the given function, which is calculated for the upper bound of integration minus lower bound of integration.

Definite Integral Properties :-

Properties Description
Property 1 p∫q f(a) da = p∫q f(t) dt
Property 2 p∫q f(a) d(a) = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0
Property 3 p∫q f(a) d(a) = p∫r f(a) d(a) + r∫q f(a) d(a)
Property 4 p∫q f(a) d(a) = p∫q f( p + q – a) d(a)
Property 5 o∫p f(a) d(a) = o∫p f(p – a) d(a)

 

Integration by Substitutions :-

Substitution method is used, when a suitable substitution of variable leads to simplification of integral.
If I = ∫f(x)dx, then by putting x = g(z), we get
I = ∫ f[g(z)] g'(z) dz
Note: Try to substitute the variable whose derivative is present in the original integral and final integral must be written in terms of the original variable of integration.

Note :-
(i) Keep in mind, ILATE is not a rule as all questions of integration by parts cannot be done by above method.
(ii) It is worth mentioning that integration by parts is not applicable to product of functions in all cases. For instance, the method does not work for ∫√x sinx dx. The reason is that there does not exist any function whose derivative is √x sinx.
(iii) Observe that while finding the integral of the second function, we did not add any constant of integration.

Exe-16(a)

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Evaluate the following integrals :

Question 1: (i) ∫ …… cot x dx

(ii) …………………… dx/ sin 2x

(iii) …………………. dx

(iv) …………………. dx

Question 2: (i) ∫ ………

………………….

Question 4:  Prove that :

(i) ∫ …………….. = π/4

(ii) ………………

Question  5: Evaluate :

……………………….

Question  9: Prove that : ……………..

Question  10:   (i) If ……. value of a

(ii) If ……………… find both and b

(iii) If f(x) is of the form ………

(iv) If ………………. Find the value of k.

Exe-16(b)

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Evaluate the following integrals :

Question 1: (i) ∫ ……  dx

(ii) …………………… dx

(iii) …………………. dx

(iv) …………………. dx

………………….

…………………..

Question 2: (i) ∫ ……  dx

(ii) …………………… dx

(iii) …………………. dx

(iv) …………………. dx

………………….

Question 3: ………………..

Question 7:  Evaluate :  ∫ ……  dx

Question 8: Prove that : ……………..

Exe-16(c)

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Question 1: (i) ∫ ……  10

(ii) …………………… 4

(iii) …………………. π/4

(iv) …………………. π/4

………………….

Evaluate the following integrals :

Question 2: ………………

Question 8: Put 2 -x = t²

Question 9: ………………

Exe-16(d)

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Evaluate the following integrals as limit of sum :

Question 1: (i) ∫ ……  dx

…………………

Question 2: ………………….

Question 4:  Prove that :

(i) ∫ ……  dx

…………………….

Question 6: (i) ∫ ……  dx

(ii) ………………

Self Revision

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Question 1:  ∫ ……  dx

Question 2:  ∫ ……  dx

Question 14: prove that ∫ ……  dx = π/4

Question 15: prove that ∫ ……  dx = 0

Question 43 : Using properties of definite integral ………………..

Question 50: Using properties of definite integrals ……………

Chapter Test

OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16

Evaluate the following integrals :

Question 1:  ∫ ……

Question 2:  ∫ ……  dx

Question 4:  ∫ ……  dx

Question 18: The value of the integral … is

Question 19: If  ∫ ……  (x)

-: End of Definite Integral S. Chand ISC Class-12 Maths Solution :-

Return to :-  OP Malhotra S. Chand ISC Class-12 Maths Solutions

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6 thoughts on “OP Malhotra Definite Integral ISC Class-12 Maths Solutions Ch-16”

  1. i am in great need of isc maths op malhotra answer key….i dont go to any coaching class…this was the only backup i had..but i see here that answers are removed….sir please help us by providing them back

    Reply

    • dear student / well wisher / Teacher
      the previous version of 2020-21 has been removed because council has decided to start new session from 1st April Therefore we are upgrading the solutions of 2021-22 editions
      Sorry for inconvenience
      thanks
      team icsehelp

      Reply

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