Question

If the substitution u=x/2 is made, the intergal from 2 to 4 1-(x/2)^2/x dx =

Answer 1

$\displaystyle\int_2^4\frac{1-\left(\frac x2\right)^2}x\,\mathrm dx$

Setting $u=\dfrac x2$ gives $\mathrm du=\dfrac{\mathrm dx}2$, or $2\,\mathrm du=\mathrm dx$. This also makes $2u=x$.

When $x\to2^+$, you have $u\to\dfrac22=1$, while when $x\to4^-$, you have $u\to\dfrac42=2$.

Putting everything together, you find that

$\displaystyle\int_2^4\frac{1-\left(\frac x2\right)^2}x\,\mathrm dx=\int_1^2\frac{1-u^2}{2u}(2\,\mathrm du)=\int_1^2\frac{1-u^2}u\,\mathrm du$

which means (B) is the correct answer.