Question

If the substitution u=(x/2) is made, the integral...

Answer 1

\[\int\limits_{2}^{4}(\frac{ 1-\frac{ x }{ 2 }^2) }{ x }dx=? \]

Answer 2

I have to clarify, the \[\frac{ x ^{2} }{ 2 } \] should be \[(\frac{ x }{ 2 })^2\]

Answer 3

\[\large\int_2^4\frac{1-\left(\frac{x}{2}\right)^2}{x}dx\;\;?\]

Answer 4

Yes

Answer 5

\[u=\frac{x}{2}\Rightarrow x=2u\\
du=\frac{1}{2}dx\\
2\;du=dx\]
\[\int_1^2\frac{1-u^2}{2u}(2\;du)=\int_1^2\frac{1-u^2}{u}\;du\]

Answer 6

If you were wondering why the limits change:
\[\text{lower: }x=2\;\Rightarrow\;u=\frac{2}{2}=1\\
\text{upper: }x=4\;\Rightarrow\;u=\frac{4}{2}=2\]

Answer 7

Ah of course! I forgot I could express x in terms of u. Thank you so much!

Answer 8

You're welcome!