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-Fraction
Rs Aggarwal Class 12 PDF (All Chapters PDF)
|Chapter no.||chapter name|
|4||Inverse Trigonometric Functions||Download|
|7||Adjoint and Inverse Of Matrix||Download|
|8||System of Linear Equations||Download|
|9||Continuity and Differentiability||Download|
|11||Applications of Derivatives||Download|
|13||Methods of integration||Download|
|14||Some Special integrals||Download|
|15||Integration using Partial Functions||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|
|30||Bayes’s Theorem and its Applications||Download|