[ QUOTE ]
S(cuberoot(x))*ln(x)dx
The
S is supposed to be the integral sign.
It seems like it should be easy, but I can't get to the correct answer -- which according to
www.calc101.com is --
(3/16)*x^(4/3)(4ln(x)-3)+C
I'm assuming I need to use integration by parts, but I keep getting to the same wrong answer. I'm dumb, please help. Also, I know the answer so obviously I need to figure out the steps to get to the answer.
Thank you.
[/ QUOTE ]
u = ln(x)
dv = x^(1/3)dx
du = (1/x)dx
v = (3/4)x^(4/3)
S(cuberoot(x))*ln(x)dx =
Su*dv = uv -
Svdu + C
= ln(x)*(3/4)x^(4/3) -
S(3/4)x^(4/3)*(1/x)dx + C
= ln(x)*(3/4)x^(4/3) - (3/4)*
Sx(1/3)dx + C
= ln(x)*(3/4)x^(4/3) - (3/4)*(3/4)x^(4/3) + C
= ln(x)*(3/4)x^(4/3) - (3/4)^2*x(4/3) + C
= (3/4)x^(4/3)*[ln(x) - 3/4] + C
divide first term by 4, and multiply second term by 4:
= (3/16)x^(4/3)*[4ln(x) - 3] + C.