Calculus help ASAP!!! Show that the function F(x) = \int\limits^(5x)_(2x) {(1)/(t)} \, dt is constant over the interval (0, +∞)

Question

Calculus help ASAP!!!

Show that the function
`F(x) = \int\limits^(5x)_(2x) {(1)/(t)} \, dt` is constant over the interval (0, +∞)

Answer 1

Let
`c\in(0,\infty)`. Then the definite integral can be split up at
`t=c` so that


`\displaystyle F(x)=\int_(t=2x)^(t=5x)\frac{\mathrm dt}t`

`\displaystyle=\int_(t=2x)^(t=c)\frac{\mathrm dt}t+\int_(t=c)^(t=5x)\frac{\mathrm dt}t`

`\displaystyle=-\int_(t=c)^(t=2x)\frac{\mathrm dt}t+\int_(t=c)^(t=5x)\frac{\mathrm dt}t`

Now take the derivative. By the fundamental theorem of calculus, you have


`\displaystyle(\mathrm dF)/(\mathrm dx)=(\mathrm d)/(\mathrm dx)\left[\int_(t=c)^(t=5x)\frac{\mathrm dt}t-\int_(t=c)^(t=2x)\frac{\mathrm dt}t\right]`

`=\displaystyle\frac1{5x}(\mathrm d)/(\mathrm dx)[5x]-\frac1{2x}(\mathrm d)/(\mathrm dx)[2x]`

`=\frac5{5x}-\frac2{2x}`

`=\frac1x-\frac1x`

`=0`

Then integrating with respect to
`x`, we recover
`F(x)` and find that


`\displaystyle\int(\mathrm dF)/(\mathrm dx)\,\mathrm dx=\int0\,\mathrm dx`

`F(x)=C`

where
`C` is an arbitrary constant.