Q. 48 - 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.)

${\int }_1^{e}2\left(\ln x\right)\left(\frac{1}{x}\right)dx$

Step-by-Step Solution

Answer

${\int }_1^{e}2\left(\ln x\right)\left(\frac{1}{x}\right)dx=1$.

Step 1. Given information.

A definite integral is given as ${\int }_1^{e}2\left(\ln x\right)\left(\frac{1}{x}\right)dx$.

Step 2. Solution.

Let

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

which means

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

Step 3. Using the Fundamental Theorem of Calculus.

We get

$$\begin{gathered} {\int }_1^{e}2\left(\ln x\right)\left(\frac{1}{x}\right)dx \\ ={\left[{\ln }^{2}x\right]}_1^e \{f\text{'}(g\left(x\right))g\text{'}(x)dx=f(g\left(x\right)\left)\right\} \\ ={\ln }^{2}e-{\ln }^{2}1 \\ =1-0 \\ =1 \end{gathered}$$

The exact value of the given definite integral is 1.

Step 4. The graph to verify the answer is

The solution is area under graph which is

$a\approx 1$