Integrals ML Aggarwal ISC Class-12 Maths Solutions

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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 CISCE 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

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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.