Question

What is the antiderivative of `ln(x)/sqrtx`?

Answer 1

`2sqrt(x)(ln(x)-2) + C`

Explanation:

For the given function, finding the antiderivative is equivalent to finding the indefinite integral. We will proceed by applying Integration by Parts.

Let `u = ln(x)` and `dv = 1/sqrt(x)dx`

Then `du = 1/xdx` and `v = 2sqrt(x)`

From the integration by parts formula `intudv = uv - intvdu`

`intln(x)/sqrt(x)dx = 2sqrt(x)ln(x) - int2sqrt(x)*1/xdx`

`= 2sqrt(x)ln(x) - 2int1/sqrt(x)dx`

`= 2sqrt(x)ln(x) - 2(2sqrt(x)) + C`

`= 2sqrt(x)(ln(x)-2) + C`