## 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. \(\int\frac{1}{x^4-1}dx\)