Q. 16 - Calculus

Question

Show by exhibiting a counterexample that, in general, $\int \frac{f\left(x\right)}{g\left(x\right)}dx\ne \frac{\int f\left(x\right)dx}{\int g\left(x\right)dx}$ . In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.

Step-by-Step Solution

Answer

The counterexample is $f\left(x\right)=x^2, g\left(x\right)=x$.

Step 1. Given Information.

The given inequality to prove is $\int \frac{f\left(x\right)}{g\left(x\right)}dx\ne \frac{\int f\left(x\right)dx}{\lg \left(x\right)dx}$.

Step 2. Conclusion.

Let the two functions be:

$f\left(x\right)=x^2,g\left(x\right)=x$

Now, we can write,

$$\begin{gathered} \int \frac{f\left(x\right)}{g\left(x\right)}dx \\ =\int \frac{x^2}{x}dx \\ =\int xdx \\ =\frac{x^2}{2}+C \end{gathered}$$

Also,

$$\begin{gathered} \frac{\int f\left(x\right)dx}{\int g\left(x\right)dx}==\frac{\int x^{2}dx}{\int xdx} \\ Therefore, \\ \int \frac{f\left(x\right)}{g\left(x\right)}dx=\frac{\frac{x^3}{3}+C}{\frac{x^2}{2}+C} \\ \ne =\frac{\int f\left(x\right)dx}{\int g\left(x\right)dx} \end{gathered}$$