Please solve in python!! Consider the following definite integral:

${\displaystyle \underset{0}{\overset{}{}}}f(x)dx,$ where $f(x)=ln(1+\frac{1}{x^k}),k2.$

Before trying to find the exact area under a curve, it is sometimes wise to numerically estimate the area first to avoid chasing after a value that is infinite or does not exist. Approximate values for integrals are also useful for empirically checking manual integration.

Set $k=2$ for parts $($a$),($b$),$ and $($c$)$ below.

$($a$)$ Use integrate.quad$()($short for quadrature$)$ from the scipy package to approximate the numerical value of the integral. Make a conjecture about the integral's exact value.

Note: You will need to write $f(x)$ as a lambda expression for use in quad$()$; for example, the function $g(x)=x^2+1$ would be written as:

$g=$ lambda $x$:$,x****2+1$

The linked documentation above contains additional details.

$($b$)$ Compute and print the antiderivative $F(x)$ using sp$.$integrate$().$ What specific integration technique$($s$)$ could you have used to find this result by hand?

$($c$)$ Use sp$.$limit$()$ and invoke the fundamental theorem of definite integration to exactly confirm your conjecture from part $($a$).$

$($d$)$ Use integrate.quad$()$ to approximate ${\displaystyle \underset{0}{\overset{}{}}}f(x)dx$ for $k=3,4,5,$ then compute each of these integrals exactly with sp$.$integrate$().$

$($e$)$ Define a symbol k with the options integer $=$ True, positive $=$ True and compute an exact formula for ${\displaystyle \underset{0}{\overset{}{}}}f(x)dx$ if $k2$ is an arbitrary integer.