Integrals ML Aggarwal ISC Class-12 (Vol-2) Understanding APC Mathematics Solutions Chapter-8. Step by step Solutions of ML Aggarwal ISC Understanding APC Mathematics Class-12 Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.
Integrals ML Aggarwal ISC Class-12 Understanding APC Mathematics Solutions Chapter-8
| Class: | 12th |
| Subject: | Mathematics |
| Chapter : | Ch-8 Integrals |
| Board | ISC Board |
| Writer | ML Aggarwal ISC Understanding ( Vol-2) |
| Publications | APC Arya Publications ( 2020-21 ) |
-: Select Topics :-
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 Understanding APC Mathematics Solutions
(page-662)
Exercise – 8.2
Integrals ML Aggarwal ISC Class-12 Understanding APC Mathematics Solutions
(page-667,668)
Exercise – 8.3
ML Aggarwal ISC Class-12 Understanding APC Mathematics Solutions
(page- 675, 676)
Exercise – 8.4
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 680, 681)
Exercise – 8.5
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 693, 694, 695)
Exercise – 8.6
ML Aggarwal Understanding APC Mathematics Solutions Integrals Class-12
(page- 699, 691)
Exercise – 8.7
ML Aggarwal Understanding APC Mathematics Solutions Integrals Class-12
(page- 709, 710)
Exercise – 8.8
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 720)
Exercise – 8.9
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 725)
Exercise – 8.10
ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 742, 743)
Exercise – 8.11
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 756, 757)
Exercise – 8.12
ML Aggarwal Understanding APC Mathematics Solutions Integrals Class-12
(page- 764, 765)
Exercise – 8.13
ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 769, 770)
Exercise – 8.14
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 774)
Exercise – 8.15
ML Aggarwal Understanding APC Mathematics Solutions Integrals Class-12
(page- 783)
Exercise – 8.16
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 795, 796)
Exercise – 8.17
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 812, 813)
Exercise – 8.18
ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 849, 850, 851)
Exercise – 8.19
ML Aggarwal ISC Understanding APC Mathematics Solutions Integrals Class-12
(page- 863, 864, 865)
Chapter Test
ML Aggarwal Understanding APC Mathematics Solutions Integrals Class-12
(page-866, 867, 868, 869 )
-: End of Integrals ML Aggarwal ISC Class-12 Solutions :-
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