Q. 55 Use the Fundamental Theorem of C... [FREE SOLUTION]

Chapter 4: Q. 55 (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.)
${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x (c s c^{2} (x^{2})) d x$

Short Answer

Expert verified

${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x (c s c^{2} (x^{2})) d x = \frac{1}{2} c o t (\frac{蟺^{2}}{16}) - \frac{1}{2} c o t (\frac{蟺^{2}}{4})$ .

Step by step solution

Step 1. Given information.

A definite integral is given as ${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x (c s c^{2} (x^{2})) d x$ .

Step 2. Using the Fundamental theorem of Calculus.

Let

$f (x) = - c o t x g (x) = x^{2}$

such that

$f' (x) = c s c^{2} x g' (x) = 2 x f' (g (x)) = c s c^{2} (x^{2})$

Now we get

localid="1648766755631" ${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x (c s c^{2} (x^{2})) d x = \frac{1}{2} {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} 2 x (c s c^{2} (x^{2})) d x = - \frac{1}{2} {[co t (x^{2})]}_{\frac{蟺}{4}}^{\frac{蟺}{2}} [f' (g (x)) g' (x) d x = f (g (x))] = - \frac{1}{2} [c o t {(\frac{蟺}{2})}^{2} - c o t {(\frac{蟺}{4})}^{2}] = \frac{1}{2} c o t (\frac{蟺^{2}}{16}) - \frac{1}{2} c o t (\frac{蟺^{2}}{4})$

The exact value of the given definite integral is $\frac{1}{2} c o t (\frac{蟺^{2}}{16}) - \frac{1}{2} c o t (\frac{蟺^{2}}{4})$ .

Step 3. The graph to verify the answer is

The solution is area under graph which is

$a 鈮� 1.330759 鈮� \frac{1}{2} [c o t (\frac{蟺^{2}}{16}) - c o t (\frac{蟺^{2}}{4})]$

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using the Fundamental theorem of Calculus.

Step 3. The graph to verify the answer is

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91影视

Use the Fundamental Theorem of Calculus to find the exactvalues of each of the definite integrals in Exercises 19鈥�64. Usea graph to check your answer. (Hint: The integrands that involveabsolute values will have to be considered piecewise.)鈭�蟺4蟺2x(csc2(x2))dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using the Fundamental theorem of Calculus.

Step 3. The graph to verify the answer is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.