# 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:

1. Write the rational function in the form of a fraction.
2. Factor the denominator of the fraction into its irreducible factors.
3. Express the numerator as the sum of a polynomial and a sum of simpler fractions.
4. Integrate each of the simpler fractions separately.
5. Integrate the polynomial.
6. 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.

1. Write the fraction in standard form: (x^2 – 1)/(x^3 – x^2 – x + 1) = (x^2 – 1)/((x-1)(x^2+x+1))
2. Factor the denominator: (x^2 – 1)/((x-1)(x^2+x+1))
3. 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)
4. 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.
5. Integrate the polynomial. x^2 dx = (x^3)/3 + C
6. 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

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