Expert answers to "Evaluate: ∫24ln⁡(9−x)dxln⁡(9−x)+ln⁡(x+3)"

Pamela Meyer

Answered question

2021-12-13

Evaluate:

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

zurilomk4

Beginner2021-12-14Added 35 answers

Let

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

Now, use that

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

Then

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

Add up these two integrals to get

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

Thus,

I = 1

In order to prove (1), write the integral using another variable, say, t:

\int_{a}^{b} f (a + b - x) d x = \int_{a}^{b} f (a + b - t) d t

In the latter one, set

x = a + b - t

and

d t = - d x

and change the limits of integration to obtain

\int_{a}^{b} f (a + b - t) d t = - \int_{b}^{a} f (x) d x

= \int_{a}^{b} f (x) d x

Mason Hall

Beginner2021-12-15Added 36 answers

Thank you.

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