Q. 53 Calculate each of the integrals ... [FREE SOLUTION]

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Chapter 5: Q. 53 (page 442)

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{\sin x}{\cos x + \cos^{2} x} d x$

Short Answer

Expert verified

The value is $- \ln | 1 - \tan^{2} \frac{x}{2} | + C$

Step by step solution

Step 1. Given Information:

Given integral : $鈭� \frac{\sin x}{\cos x + \cos^{2} x} d x$

We want to calculate each of the integrals.

Step 2. Calculation:

$鈭� \frac{\sin x}{\cos x + \cos^{2} x} d x = 鈭� \frac{\frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}}}{\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} + {(\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}})}^{2}} d x U s e S u b s t i t u t i o n t = \tan \frac{x}{2} A l s o d x = \frac{1}{t} d t a n d t = \frac{1}{2} (1 + \tan^{2} \frac{x}{2}) 鈭� \frac{\frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}}}{\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} + {(\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}})}^{2}} 脳 \frac{1}{\frac{1}{2} (1 + \tan^{2} \frac{x}{2})} d t S u b s t i t u t e t = \tan \frac{x}{2} 鈭� \frac{\frac{2 t}{1 + t^{2}}}{\frac{1 - t^{2}}{1 + t^{2}} + {(\frac{1 - t^{2}}{1 + t^{2}})}^{2}} 脳 \frac{1}{\frac{1}{2} (1 + t^{2})} d t = 鈭� \frac{\frac{2 t}{1 + t^{2}}}{\frac{(1 + t^{2}) (1 - t^{2}) + {(1 - t^{2})}^{2}}{{(1 + t^{2})}^{2}}} 脳 \frac{2}{1 + t^{2}} d t = 鈭� \frac{\frac{2 t}{1 + t^{2}}}{\frac{(1 - t^{4}) + {(1 - t^{2})}^{2}}{{(1 + t^{2})}^{2}}} 脳 \frac{2}{1 + t^{2}} d t = 鈭� \frac{2 t (1 + t^{2})}{1 - t^{4} + {(1 - t^{2})}^{2}} 脳 \frac{2}{1 + t^{2}} d t = 鈭� \frac{4 t}{1 - t^{4} + {(1 - t^{2})}^{2}} d t = 鈭� \frac{4 t}{1 - t^{4} + (1 - 2 t^{2} + t^{4})} d t = 鈭� \frac{4 t}{2 - 2 t^{2}} d t = 鈭� \frac{2 t}{1 - t^{2}} d t = - \ln | 1 - t^{2} | + C S u b s t i t u t e b a c k t = \tan \frac{x}{2} = - \ln | 1 - \tan^{2} \frac{x}{2} | + C$

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Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{\sin x}{\cos x + \cos^{2} x} d x$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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91影视

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.鈭�sinxcosx+cos2xdx

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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Most popular questions from this chapter

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Geometry

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{\sin x}{\cos x + \cos^{2} x} d x$