Q. 52 - Calculus

Question

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises 19–64. Use

a graph to check your answer. (Hint: The integrands that involve

absolute values will have to be considered piecewise.)

${\int }_2^3\frac{\ln x}{x}dx$

Step-by-Step Solution

Answer

${\int }_2^3\frac{\ln x}{x}dx=\frac{1}{2}[{\ln }^{2}3-{\ln }^{2}2]$.

Step 1. Given information.

A definite integral is given as ${\int }_2^3\frac{\ln x}{x}dx$.

Step 2. Using the Fundamental theorem of Calculus.

Let

$$\begin{gathered} f\left(x\right)=x^2 \\ g\left(x\right)=\ln x \end{gathered}$$

then

$$\begin{gathered} f\text{'}\left(x\right)=2x \\ g\text{'}\left(x\right)=\frac{1}{x} \\ f\text{'}\left(g\right(x\left)\right)=2\left(\ln x\right) \end{gathered}$$

Now we get

$$\begin{gathered} {\int }_2^3\frac{\ln x}{x}dx \\ =\frac{1}{2}{\int }_2^3\frac{2\ln x}{x}dx \\ =\frac{1}{2}{\left[{\ln }^{2}x\right]}_2^3 [f\text{'}(g\left(x\right))g\text{'}(x)dx=f(g\left(x\right)\left)\right] \\ =\frac{1}{2}[{\ln }^{2}3-{\ln }^{2}2] \end{gathered}$$

The exact value of the given definite integral is $\frac{1}{2}[{\ln }^{2}3-{\ln }^{2}2]$.

Step 3. The graph to verify the answer is

The solution is area under graph which is

$$\begin{gathered} a=0.36324797 \\ \approx \frac{1}{2}[{\ln }^2\left(3\right)-{\ln }^2\left(2\right)] \end{gathered}$$