Help on evaluating an integral

[Bradyams]

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.

[BruceZ]

[ 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.

[BruceZ]

[ 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 = Sudv

= uv - Svdu

= ln(x)*(3/4)x^(4/3) - S(3/4)x^(4/3)*(1/x)dx

= ln(x)*(3/4)x^(4/3) - (3/4)*Sx^(1/3)dx

= ln(x)*(3/4)x^(4/3) - (3/4)*(3/4)x^(4/3) + C

= (3/4)x^(4/3)*[ln(x) - 3/4] + C

To get calc101 answer, divide first term by 4, and multiply second term by 4:

= (3/16)x^(4/3)*[4ln(x) - 3] + C.