Chapter 4: Q. 42 (page 373)

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
$\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x$

Short Answer

Expert verified

Ans: The exact value of, $\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x = \frac{5 (2 (\ln (2) + \ln (- 1)) - 2 \ln (- 3) - 1)}{4}$

Step by step solution

Step 1. Given information.

given expression,

$\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x$

Step 2. The objective is to determine the exact value of the definite integral.

The exact value is calculated as shown below,

$\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x = \frac{1}{2} \int_{- 3}^{2} x d x + \frac{5}{2} \int_{- 3}^{2} \frac{1}{x} d x = [\frac{x^{2}}{4} + \frac{5 \ln (x)}{2}]_{- 3}^{2} = [\frac{5 \ln (| x |) + \frac{x^{2}}{2}}{2}]_{- 3}^{2} = \frac{5 \ln (2) + 5 \ln (- 1) - 5 \ln (- 3) - \frac{5}{2}}{2} = \frac{5 (2 (\ln (2) + \ln (- 1)) - 2 \ln (- 3) - 1)}{4}$

Therefore, the exact value is $\frac{5 (2 (\ln (2) + \ln (- 1)) - 2 \ln (- 3) - 1)}{4}$

Step 3. Check:

The required graph is,

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Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
$\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine the exact value of the definite integral.

Step 3. Check:

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Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer. ∫−32 x2+52xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine the exact value of the definite integral.

Step 3. Check:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Study anywhere. Anytime. Across all devices.

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
$\int_{- 3}^{2} \frac{x^{2} + 5}{2 x} d x$