Chapter 4: Q. 36 (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.
$\int_{0}^{1} \frac{x}{1 + x^{2}} d x$

Short Answer

Expert verified

Ans: The exact value of, $\int_{0}^{1} \frac{x}{1 + x^{2}} d x = \frac{1}{2} \ln | 2 | .$

Step by step solution

Step 1. Given information.

given,

$\int_{0}^{1} \frac{x}{1 + x^{2}} d x$

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

Let,

$\begin{aligned} u = 1 + x^{2} \\ d u = 2 x d x \end{aligned}$

The exact value is calculated as shown below,

$\begin{aligned} \int_{0}^{1} \frac{x}{1 + x^{2}} d x \\ = \int_{0}^{1} \frac{1}{2 u} d u \\ = \frac{1}{2} \int_{0}^{1} \frac{1}{u} d u \\ = \frac{1}{2} [\ln (| u |)]_{0}^{1} \\ = \frac{1}{2} [\ln (|1 + 1^{2}|) - \ln (|1 + 0^{2}|)] \\ = \frac{1}{2} (\ln | 2 | - \ln | 1 |) \\ = \frac{1}{2} \ln | 2 | \end{aligned}$

Therefore, the exact value is $\frac{1}{2} \ln | 2 |$ .

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.
$\int_{0}^{1} \frac{x}{1 + x^{2}} d x$

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. ∫01 x1+x2dx

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

Logic and Functions

Geometry

Applied Mathematics

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Statistics

Discrete Mathematics

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.
$\int_{0}^{1} \frac{x}{1 + x^{2}} d x$