Have a question on how to do an integral.

[PCP]

How do you do this integral?

Integral of ln(x)/x^(1/2)dx

 

[LeonhardEuler]

If you re-write it as $x^{-\frac{1}{2}}\ln(x)$ integration by parts comes to mind. It will work.

 

[quicknote]

To solve this integral, you can use the substitution method. Let u = √x, then du = (1/2x^(1/2))dx. Substituting this into the integral, we get ∫ln(u)/u du. We can then use integration by parts, with u = ln(u) and dv = 1/u du. This gives us the integral ∫ln(u)du = uln(u) - ∫1du. Substituting back in for u, we get √xln(√x) - ∫1du = √xln(√x) - u + C. Finally, substituting back in for u, we get the final answer of √xln(√x) - √x + C.