Question

Evaluate the integral: \(\int {\frac{{axdx}}{{{x^2} - bx}}} \)

Short answer

Hence, the value of \(\int {\frac{{axdx}}{{{x^2} - b}}} \) is \(a\log \left| {x - b} \right| + c\)

Step-by-step solution

Factorizing

\(\int {\frac{{axdx}}{{{x^2} - bx}}} = \int {\frac{{axdx}}{{x(x - b)}} = \int {\frac{{adx}}{{x - b}}} } \)

Finding value

\(\int {\frac{{adx}}{{x - b}}} \)

Let \(x - b = v\)

\(\begin{array}{l}1 = \frac{{dv}}{{dx}}\\ \Rightarrow dx = dv\\ \Rightarrow a\int {\frac{{dv}}{v} = a\log \left| v \right|} + c\\ = a\log \left| {x - b} \right| + c\end{array}\)

Hence, the value of \(\int {\frac{{axdx}}{{{x^2} - b}}} \) is \(a\log \left| {x - b} \right| + c\)