Q. 34 Use the Second Fundamental Theor... [FREE SOLUTION]

Chapter 4: Q. 34 (page 399)

Use the Second Fundamental Theorem of Calculus to write down three antiderivatives of each function fin Exercises 31鈥�34.
$f (x) = \frac{\ln (\sin x)}{x^{2} - 1}$

Short Answer

Expert verified

The three antiderivatives for the function $f (x) = \frac{\ln (\sin x)}{x^{2} - 1}$ are $F (x) = {鈭�}_{0}^{x} \frac{\ln (\sin t)}{t^{2} - 1}, F (x) = {鈭�}_{- 4}^{x} \frac{\ln (\sin t)}{t^{2} - 1}, F (x) = {鈭�}_{2}^{x} \frac{\ln (\sin t)}{t^{2} - 1}$ .

Step by step solution

Step 1. Given Information.

The function:
$f (x) = \frac{\ln (\sin x)}{x^{2} - 1}$

Step 2. Graph the function.

Graph the function.

Step 3. Find the anti-derivatives.

If F is an anti-derivative of f,f is continuous on [a,b], then $F (x) = {鈭�}_{a}^{x} f (t) . d t$ for all $x 鈭� [a, b]$ .

So, $F (x) = {鈭�}_{0}^{x} \frac{\ln (\sin t)}{t^{2} - 1}$ is defined to be an anti-derivative of the given function.

Similarly, the other two anti-derivatives are:

$F (x) = {鈭�}_{- 4}^{x} \frac{\ln (\sin t)}{t^{2} - 1}, F (x) = {鈭�}_{2}^{x} \frac{\ln (\sin t)}{t^{2} - 1}$

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91影视

Use the Second Fundamental Theorem of Calculus to write down three antiderivatives of each function fin Exercises 31鈥�34.
$f (x) = \frac{\ln (\sin x)}{x^{2} - 1}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Graph the function.

Step 3. Find the anti-derivatives.

One App. One Place for Learning.

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91影视

Use the Second Fundamental Theorem of Calculus to write down three antiderivatives of each function fin Exercises 31鈥�34.f(x)=ln(sinx)x2-1

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Graph the function.

Step 3. Find the anti-derivatives.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.

Use the Second Fundamental Theorem of Calculus to write down three antiderivatives of each function fin Exercises 31鈥�34.
$f (x) = \frac{\ln (\sin x)}{x^{2} - 1}$