/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 Evaluate \(\int_{2}^{4} \frac{\s... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 100

Evaluate \(\int_{2}^{4} \frac{\sqrt{\ln (9-x)} d x}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}\)

Short Answer

Expert verified
The value of the integral \( \int_{2}^{4} \frac{\sqrt{\ln (9-x)} d x}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}} \) is \( 2 + 3\ln(7)-3\ln(5) \)

Step by step solution

01

Simplify the integrand

Start by observing the terms under the square root are exactly the same. The denominator contains these terms, but with one term containing plus in between and the other not. Therefore, we can simplify the integrand using this logic: \( \frac{\sqrt{\ln (9-x)}}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}} = 1 - \frac{\sqrt{\ln (x+3)}}{\sqrt{\ln (9-x)} + \sqrt{\ln (x+3)}} \)
02

Change of variables

Next, a change of variables will simplify the expression. Letting \( u = \ln (x+3) \) and \( v = \ln (9-x) \), so that \( du= \frac{dx}{x+3} \) and \( dv=-\frac{dx}{9-x} \). But \( du + dv = 0 \), then \( dx = (x + 3) du \) and \( dx = -(9 - x) dv \). By substituting \( v = \ln (9-x) \) into the first equation, we get \( dx = (x + 3) dv \).
03

Substitute the variables in the integral

The original integral can now be expressed in terms of u only resulting in: \[ \int_{2}^{4} (1 - \frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}}) (x + 3) dv \]
04

Separate the integral

Separate the integral into two parts: \[ \int_{2}^{4} (x + 3) dv - \int_{2}^{4} \frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}} (x + 3) dv \]. Notice that the first part of the integral equals to zero since \( dv = - du \). As for second part, the denominator can be simplified to give \( \int_{2}^{4} (x + 3) dv = \int_{\ln{5}}^{\ln{7}} (e^u + 3) du = e^{\ln{7}} - e^{\ln{5}} + 3 (\ln{7} - \ln{5}) \).
05

Calculate the definite integral

From the simplification of the first integral, integrate the remaining portion. Therefore, the result of the integral is: \( 7 - 5 + 3 (\ln{7} - \ln{5}) = 2 + 3 \ln{\frac{7}{5}} = 2+ 3[\ln{7} - \ln{5}] = 2+3\ln(7)-3\ln(5)\)

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