Integrals ML Aggarwal ISC Class-12 Maths Solutions

Integrals ML Aggarwal ISC Class-12 Maths Solutions Chapter-8. Step by step Solutions of ML Aggarwal ISC Mathematics for Class-12 Questions with Chapter Test. Visit official Website for detail information about ISC Board Class-12 Mathematics.

Integrals ML Aggarwal ISC Class-12 Maths Solutions

 Board ISC Class 12 Subject Mathematics Chapter-8 Integrals Session 2024-25 Topics Solutions of ML Aggarwal

Definition of integral

Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given

Concepts on Integrals

• Integration as an inverse process of differentiation
• Geometrical interpretation of indefinite integral
• Some properties of indefinite integrals
• Comparison between differentiation and integration
• Methods of integration
• Integration by substitution
• Integration using trigonometric identities
• Integrals for some particular functions
• Integration by partial fractions
• Integration by parts
• Integral of the type
• Integrals for some more types
• Definite integral
• Definite integral as a limit of a sum
• The fundamental theorem of calculus
• Area function
• The first fundamental theorem of integral calculus
• The second fundamental theorem of integral calculus
• Evaluation 0f definite integrals by substitution
• Some properties of definite integrals

Integral Types

The integral calculus is of the two forms, namely

(i) Indefinite Integral

(ii) Definite Integral

Anti derivative Functions

Let us have a look at various anti derivative of functions.

Some indefinite integrals which can be evaluated by direct substitutions:

• 1) If integral is of the form ∫ f(g(x)) g'(x) dx, then put g(x) = t, provided ∫ f(t) exists.
• 2) ∫ f'(x)/f(x) dx = ln |f (x)| + c,  By putting f (x) = t => f’ (x) dx = dt

=> ∫ dt/t = ln |t| + c = ln |f (x)| + c.

• 3) ∫ f'(x)√f(x) dx = 2 √f(x)+c,  Put f (x) = t

Then ∫ dt/√t = 2√t + c = 2√f(x) + c.

Some standard substitutions:

1) For terms of the form x2 + a2 or √x2 + a2, put x = a tanθ or a cotθ

2) For terms of the form x2 – a2 or √x2 – a2 , put x = a sec θ or a cosecθ

3) For terms of the form a2 – x2 or √x2 + a2, put x = a sin θ or a cosθ

4) If both √a+x, √a–x, are present, then put x = a cos θ.

5) For the form √(x–a)(b–x), put x = a cos2θ + b sin2θ

6) For the type (√x2+a2±x)n or (x±√x2–a2)n, put the expression within the bracket = t.

7) For 1/(x+a)n1 (x+b)n2, where n1,n2 ∈ N (and > 1), again put (x + a) = t (x + b)

• If the integrand is of the form f(x)g(x), where g(x) is a function of the integral of f(x), then put integral of f(x) = t.
• The integral of product of two functions of x is evaluated with the help of integration by parts. Let u and v be two functions of x, then ∫uv dx = u∫v dx – ∫[du/dx ∫v dx]dx
• While carrying out integration by parts, whether a function is u or v should be decided according to ILATEmethod of integration (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent).
• If both the functions are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable.
• If in the product of the two functions, one of the functions is not directly integrable like lnx, sin-1x, cos-1x, tan-1x etc. then we take it as the first function and the remaining function is taken as the second function.
• If there is no second function available, then unity is taken as the second function e.g. in the integration of∫tan-1x dx, tan-1x is taken as the first function and 1 as the second function.

Exercise – 8.1

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

Exercise – 8.2

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page-667,668)

Exercise – 8.3

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 675, 676)

(page- 680, 681)

Exercise – 8.5

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 693, 694, 695)

Exercise – 8.6

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 699, 691)

Exercise – 8.7

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 709, 710)

Exercise – 8.8

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 720)

Exercise – 8.9

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 725)

Exercise – 8.10

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 742, 743)

Exercise – 8.11

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 756, 757)

Exercise – 8.12

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 764, 765)

Exercise – 8.13

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 769, 770)

Exercise – 8.14

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 774)

Exercise – 8.15

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 783)

Exercise – 8.16

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 795, 796)

Exercise – 8.17

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 812, 813)

Exercise – 8.18

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 849, 850, 851)

Exercise – 8.19

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page- 863, 864, 865)

Chapter Test

Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(page-866, 867, 868, 869 )

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