## Part - 3 Lecture - 4 Chapter 7 Integrals

“Don’t Let Yesterday Take Up Too Much Of Today.” – Will Rogers

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INTEGRATION OF ALGEBRAIC FUNCTIONS

There are three methods to find the integral of Algebraic functions based on the type of Algebraic function.

FIRST METHOD

The first method is Partial Fractions, used only if the following conditions are satisfying:
• There is no root in either numerator or denominator (Only for rational functions)
• The denominator can be factorised
• It must be a proper fraction. In case of improper fraction, first, convert it into a proper fraction by division method.

 Form of the rational function Form of partial fraction $$\frac{px+q}{(x-a)(x-b)…(x-z)}$$ $$\frac{A}{(x-a)}+\frac{B}{(x-b)} + … +\frac{Z}{(x-z)}$$ $$\frac{px+q}{(x-a)^n}$$ $$\frac{A}{(x-a)}+\frac{B}{(x-a)^2} + … + \frac{Z}{(x-a)^n}$$ $$\frac{px^2+qx+r}{(x-a)(x^2+bx+c)}$$ $$\frac{A}{(x-a)} + \frac{Bx+C}{(x^2+bx+c)}$$Where $$x^2+bx+c$$ cannot be factorised further…

In this lecture I am discussing following questions and the last case of partial fractions

3. $$\int\frac{x^2+1}{x^2-5x+6}dx$$

4. $$\int \frac{x^2+x+1}{(x+2)(x^2+1)}dx$$

5. $$\int\frac{2x-3}{(x^2-1)(2x+3)}dx$$

6. $$\int\frac{1}{x^4-1}dx$$