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Chapter 4: Q. 52 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19鈥�64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
${鈭�}_{2}^{3} \frac{\ln x}{x} d x$

Short Answer

Expert verified

${鈭�}_{2}^{3} \frac{\ln x}{x} d x = \frac{1}{2} [\ln^{2} 3 - \ln^{2} 2]$ .

Step by step solution

01

Step 1. Given information.

A definite integral is given as ${鈭�}_{2}^{3} \frac{\ln x}{x} d x$ .

02

Step 2. Using the Fundamental theorem of Calculus.

Let

$f (x) = x^{2} g (x) = \ln x$

then

$f' (x) = 2 x g' (x) = \frac{1}{x} f' (g (x)) = 2 (\ln x)$

Now we get

${鈭�}_{2}^{3} \frac{\ln x}{x} d x = \frac{1}{2} {鈭�}_{2}^{3} \frac{2 \ln x}{x} d x = \frac{1}{2} {[\ln^{2} x]}_{2}^{3} [f' (g (x)) g' (x) d x = f (g (x))] = \frac{1}{2} [\ln^{2} 3 - \ln^{2} 2]$

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

03

Step 3. The graph to verify the answer is

The solution is area under graph which is

$a = 0.36324797 鈮� \frac{1}{2} [\ln^{2} (3) - \ln^{2} (2)]$

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Most popular questions from this chapter

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{2}^{4} (x^{2} + 1) d x$

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{6}^{1} (3 (1 - 2 x)^{2} + 4 x) d x$

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ and ${鈭�}_{3}^{6} g (x) d x = 3$ ,then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{3}^{6} f (x) d x$ .

Fill in each of the blanks:

(a) $鈭� d x = x^{6} + C .$

(b) $x^{6}$ is an antiderivative of .

(c) The derivative of $x^{6}$ is .

If f is negative on [鈭�3, 2], is the definite integral ${鈭�}_{- 3}^{2} f (x) d x$ positive or negative? What about the definite integral 鈭� ${鈭�}_{- 3}^{2} f (x) d x$ ?

See all solutions

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