Chapter 5: Q 77. (page 429)

Solve the definite integral.
${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x c s c^{2} x d x$

Short Answer

Expert verified

The solution is $\frac{1}{4} (蟺 + 2 \log 2)$ .

Step by step solution

Step 1. Given information.

The given integral is ${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x c s c^{2} x d x$ .

Step 2. First, solve the indefinite integral.

$\begin{array}{rcl} I & = & 鈭� x c s c^{2} x d x \\ = & x 鈭� c s c^{2} x d x - 鈭� [\frac{d}{d x} (x) 鈭� c s c^{2} x d x] d x \\ = & x (- c o t x) - 鈭� 1 (- c o t x) d x \\ = & - x c o t x + 鈭� c o t x d x \\ = & - x c o t x + \log |\sin x| 鈥� 鈥� 鈥� 鈥� \{鈭� c o t x d x = \log |\sin x|\} \end{array}$

Step 3. Now apply the limit of integration.

$\begin{array}{rcl} {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x c s c^{2} x d x & = & {[- x c o t x + \log |\sin x|]}_{\frac{蟺}{4}}^{\frac{蟺}{2}} \\ = & [- (\frac{蟺}{2}) c o t (\frac{蟺}{2}) + \log |\sin (\frac{蟺}{2})|] - [- (\frac{蟺}{4}) c o t (\frac{蟺}{4}) + \log |\sin (\frac{蟺}{4})|] \\ = & \frac{1}{4} (蟺 + 2 \log 2) \end{array}$

Step 4. Simplified answer.

Hence, the required value is $\frac{1}{4} (蟺 + 2 \log 2)$ .

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Solve the definite integral.
${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x c s c^{2} x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Now apply the limit of integration.

Step 4. Simplified answer.

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91影视

Solve the definite integral.鈭�蟺4蟺2xcsc2xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Now apply the limit of integration.

Step 4. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Solve the definite integral.
${鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} x c s c^{2} x d x$