Question

Definite integration

Answer 1

This should help @mathmath333
http://mathworld.wolfram.com/DefiniteIntegral.html

~AND CONGRATZ 90

Answer 2

\(\large \color{black }{\begin{align}
\int^4_2 \dfrac{\log x^2}{\log x^2+\log (36-12x+x^2)}\,dx \hspace{.33em}\\~\\
\end{align}}\)

Answer 3

What is your question?

Answer 4

\[x^2 - 12x + 36 = (x-6)^2\]

Answer 5

.

Answer 6

log = natural log, in this case or log base 10?

Answer 7

it is not specifically given whether it is a natural log or base 10

options should help


\(\large \color{black }{\begin{align}
&a.)6\hspace{.33em}\\~\\
&b.)2\hspace{.33em}\\~\\
&c.)4\hspace{.33em}\\~\\
&d.)1\hspace{.33em}\\~\\
\end{align}}\)

Answer 8

\[\large \begin{align}
I&= \int_2^4 \dfrac{\log x^2}{ \log x^2 + \log (36-12x+x^2)}\\~\\
&= \int_2^4 \dfrac{\log x}{ \log (x(6-x))}~~~~\color{red}{\star}\\~\\
&= \int_2^4 \dfrac{\log (6-x)}{ \log ((6-x)x)}~~~~\color{red}{\clubsuit }\\~\\
\end{align}\]

\(\large \color{red}{\star}+ \color{red}{\clubsuit } = 2I =\int_2^4 1 dx = 2 \\\implies I = 1 \)

Answer 9

how do you get the equality for the clover?

Answer 10

ahh skipped many steps in between, that clover step comes from definite integral property :\[\int_a^b f(x) dx = \int_a^b f(a+b-x) dx\]

Answer 11

clever!

Answer 12

what is the name of that property

Answer 13

easy to prove

Answer 14

do a u sub: u = a + b - x

if int of f(x) = F(x) you'll see that it's true

Answer 15

omg yeah that will do !!

Answer 16

actually that's a good problem for students... have them show that the property is true by utilizing u sub.