Rs Aggarwal Class 12 Integration using Partial Functions PDF
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Integration using partial fractions is a method used to evaluate integrals of rational functions, which are functions in the form of a fraction of polynomials. The method of partial fractions is based on the fact that any rational function can be expressed as the sum of a polynomial and a sum of simpler fractions called partial fractions.
What are Partial Functions in Integrals?
The general process of using partial fractions to evaluate an integral is as follows:
- Write the rational function in the form of a fraction.
- Factor the denominator of the fraction into its irreducible factors.
- Express the numerator as the sum of a polynomial and a sum of simpler fractions.
- Integrate each of the simpler fractions separately.
- Integrate the polynomial.
- Add the results of steps 4 and 5 to obtain the antiderivative of the original function.
For example, consider the integral of (x^2 – 1)/(x^3 – x^2 – x + 1)dx.
- Write the fraction in standard form: (x^2 – 1)/(x^3 – x^2 – x + 1) = (x^2 – 1)/((x-1)(x^2+x+1))
- Factor the denominator: (x^2 – 1)/((x-1)(x^2+x+1))
- Express the numerator as the sum of a polynomial and a sum of simpler fractions. (x^2 – 1) = A(x-1) + B(x^2+x+1)
- Integrate each of the simpler fractions separately. A(x-1) dx = A x dx – A dx and B(x^2+x+1) dx = Bx^2 dx + Bx dx + B dx.
- Integrate the polynomial. x^2 dx = (x^3)/3 + C
- Add the results of steps 4 and 5 to obtain the antiderivative of the original function. (A x dx – A dx) + (Bx^2 dx + Bx dx + B dx) + (x^3)/3 + C
By doing this process one can find the antiderivative of the function without knowing the original function.
It’s important to note that this method can be a bit tricky and may require the use of algebraic manipulation, especially when the denominator of the function is not easily factorable.
Ch-15-Integration-Using-Partial-FractionRs Aggarwal Class 12 PDF (All Chapters PDF)
| Chapter no. | chapter name | |
|---|---|---|
| 1 | Relations | Download |
| 2 | Functions | Download |
| 3 | Binary Operations | Download |
| 4 | Inverse Trigonometric Functions | Download |
| 5 | Matrices | Download |
| 6 | Determinants | Download |
| 7 | Adjoint and Inverse Of Matrix | Download |
| 8 | System of Linear Equations | Download |
| 9 | Continuity and Differentiability | Download |
| 10 | Differentiations | Download |
| 11 | Applications of Derivatives | Download |
| 12 | Indefinite Integrals | Download |
| 13 | Methods of integration | Download |
| 14 | Some Special integrals | Download |
| 15 | Integration using Partial Functions | Download |
| 16 | Definite integrals | Download |
| 17 | Area of Bounded Regions | Download |
| 18 | Differential equations and their formations | Download |
| 19 | Differential equations with variable separable | Download |
| 20 | Homogeneous Integral equations | Download |
| 21 | Linear Differential equations | Download |
| 22 | Vectors And Their Properties | Download |
| 23 | Scalar, or Dot product of Vectors | Download |
| 24 | Cross or vector products of Vector | Download |
| 25 | Product of three vectors | Download |
| 26 | Fundamental Concepts of 3D Geometry | Download |
| 27 | Straight Line in Space | Download |
| 28 | The Plane | Download |
| 29 | Probability | Download |
| 30 | Bayes’s Theorem and its Applications | Download |
| 31 | Probability Distribution | Download |
| 32 | Binomial Distributions | Download |
| 33 | Linear Programming | Download |