Indefinite Integrals
14.0 Reduction formulae
14.0 Reduction formulae
To solve these type of integrals, following methods are used:
If ${a{x^2} + bx + c}$ is converted to the form of perfect square or can be factorized, then standard results can be applied or solved using partial fraction method.
If ${a{x^2} + bx + c}$ can not be factorized, then it is expressed as the sum or difference of two squares by completing the square method.
Both the methods are explained with the help of example.
Question 13. Evaluate $${\int {\frac{{dx}}{{\sqrt {{x^2} - 2x + 3} }}} }$$
Solution: $$\begin{equation} \begin{aligned} I = \int {\frac{{dx}}{{\sqrt {{x^2} - 2x + 3} }}} \\ I = \int {\frac{{dx}}{{\sqrt {{x^2} - 2x + 1 - 1 + 3} }}} \\ I = \int {\frac{{dx}}{{\sqrt {{{(x - 1)}^2} + {{\left( {\sqrt 2 } \right)}^2}} }}} \sim \int {\frac{{dX}}{{\sqrt {{X^2} + {a^2}} }}} \\ I = \log \left| {(x - 1) + \sqrt {{{\left( {x - 1} \right)}^2} + {{\left( {\sqrt 2 } \right)}^2}} } \right| \\ I = \log \left| {(x - 1) + \sqrt {{x^2} - 2x + 3} } \right| + C \\\end{aligned} \end{equation} $$
