OP Malhotra Indefinite Integral-1 Standard Forms ISC Class-12 Maths Solutions Ch-13

OP Malhotra Indefinite Integral-1 Standard Forms ISC Class-12 Maths Solutions Ch-13. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-13(a), Exe-13(b. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

OP Malhotra Indefinite Integral-1 Standard  Forms ISC Class-12 Maths Solutions Ch-13

Class:	12th
Subject:	Mathematics
Chapter  :	Ch-13  Indefinite Integral-1 Standard  Forms of Section -A

Board	ISC
Writer	OP Malhotra, SK Gupta, Anubhuti Gangal
Publications	S.Chand Publications 2020-21

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-: Included Topics :- 

Exe-13(a)

Exe-13(b)

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OP Malhotra Indefinite Integral-1 Standard  Forms ISC Class-12 Maths Solutions Ch-13

Indefinite Integral :

Integration is the inverse process of differentiation.

Properties of Indefinite Integral :-

(i) ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
(ii) For any real number k, ∫k f(x) dx = k∫f(x)dx.
(iii) In general, if f₁, f₂,………, f_n are functions and k₁, k₂,…, k_n are real numbers, then
∫[k₁f₁(x) + k₂ f₂(x)+…+ k_nf_n(x)] dx = k₁ ∫f₁(x) dx + k₂ ∫ f₂(x) dx+…+ k_n ∫f_n(x) dx

Properties of  Indefinite Integral-1 

• 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.

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Exe-13(a)

OP Malhotra Indefinite Integral-1 Standard  Forms ISC Class-12 Maths Solutions Ch-13

Integrate the following function :

Question 1:

(i) sin 2x

(ii) 2 in 3x

(iii)………….

(iv)………..

(v)………….

(vi)………………

(vii)………….

(viii)………………

Question 2:

(i) cos (5 – 3x )

(ii)………..

(iii)………….

(iv)………..

(v)………….

(vi)………………

Question 3:

………………………

……………………….

……………………..

Question 14:

If f'(x)  = a sin x + b cos x and …………………

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Exe-13(b)

OP Malhotra Indefinite Integral-1 Standard  Forms ISC Class-12 Maths Solutions Ch-13

Evaluate the following integral :

Question 1:

………………..

………………

Question 20:

…………………

-: End of Indefinite Integral-1 Standard  Forms  S. Chand ISC Class-12 Maths Solution :-

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

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