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**

#### 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 x^{2} + a^{2} or √x^{2} + a^{2}, put x = a tanθ or a cotθ

2) For terms of the form x^{2} – a^{2} or √x^{2} – a^{2} , put x = a sec θ or a cosecθ

3) For terms of the form a^{2} – x^{2} or √x^{2} + a^{2}, 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 cos^{2}θ + b sin^{2}θ

6) For the type (√x^{2}+a^{2}±x)^{n} or (x±√x^{2}–a^{2})^{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
^{-1}x, cos^{-1}x, tan^{-1}x 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
^{-1}x dx, tan^{-1}x 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)

**Exercise – 8.4 **

### Integrals ML Aggarwal ISC Class-12 Mathematics Solutions

(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 )

-: End of Integrals ML Aggarwal ISC Class-12 Mathematics Solutions :-

Return to :- ML Aggarwal ISC Class 12 Maths Vol-2 Solutions

Thanks

Please share with your friends

Noice

LoL