Show that the function F of x equals the integral from 2 times x to 5 times x of 1 over t dt is constant on the interval (0, +∞).

Question

Show that the function  F of x equals the integral from 2 times x to 5 times x of 1 over t dt is constant on the interval (0, +∞).

erinkb99 · Answer

attachment

zepdrix · Answer

Hmm this seems too simple...
Can't we just integrate and show that F(x)=a number ?$$\large\rm F(x)=\int\limits_{2x}^{5x}\frac{1}{t}dt\quad=\ln|t|_{2x}^{5x}$$1/t integrates to log t, ya?

erinkb99 · Answer

yea, ln t

zepdrix · Answer

Evaluating at the limits,$$\large\rm F(x)=\ln|5x|-\ln|2x|$$

zepdrix · Answer

Do you remember your log rules? :)

erinkb99 · Answer

no

zepdrix · Answer

Well this particular rule will come in very handy,$$\large\rm \log(a)-\log(b)=\log\left(\frac{a}{b}\right)$$Understand how we can apply that?

erinkb99 · Answer

ln|5x|-ln|2x| = ln|5x/2x|

zepdrix · Answer

Good, now simplify.

erinkb99 · Answer

ln|5/2|

zepdrix · Answer

Yayyyyy we've shown that F(x) is constant! :)$$\large\rm F(x)=\ln\frac52$$

zepdrix · Answer

No x's anywhere in sight, it's just a number left over.

erinkb99 · Answer

Thanks!! I have 1 more question I need help with