Question

`\int\limits^e_1 {\frac{1}{x\sqrt{1-(logx)^(2) } } } \, dx`

Answer 1

For the integral

`I=\displaystyle\int_1^3(\mathrm dx)/(x√(1-(\log x)^2))`

(assuming
`\log x` is the natural logarithm with base
`e`) substitute
`u=\log x` and
`\mathrm du=\frac{\mathrm dx}x`. Then the integral is equivalent to

`I=\displaystyle\int_(\log1)^(\log e)(\mathrm du)/(√(1-u^2))=\int_0^1(\mathrm du)/(√(1-u^2))`

Next, substitute
`u=\sin t` and
`\mathrm du=\cos t\,\mathrm dt`:

`I=\displaystyle\int_(\sin^(-1)0)^{\sin^(-1)1}(\cos t)/(√(1-\sin^2t))\,\mathrm dt=\int_0^(\frac\pi2)(\cos t)/(√(\cos^2t))\,\mathrm dt`

We have
`√(x^2)=|x|` for all
`x`, but in the given integration interval,
`\cos t\ge0`, so

`I=\displaystyle\int_0^(\frac\pi2)(\cos t)/(\cos t)\,\mathrm dt=\int_0^(\frac\pi2)\mathrm dt=\boxed{\frac\pi2}`

(Of course, with a little foresight, you could have immediately combined the two substitutions and started off with letting
`\sin u=\log x`.)