what is the integral of (x^4(ln x) dx)?

Question

this is integration by parts. My main problem is finding the anti-derivative of (1/5)x^4....

so confused, if you can help, please explain or show your work.

Thanks!

Answer 1

Integrate ∫(x^4(ln x) dx).
Let
u = lnx and dv = x^4 dx
du = dx/x and v = x^5/5

∫(x^4(ln x) dx) = (1/5)(x^5)lnx - ∫(x^5/5)(dx/x)
= (1/5)(x^5)lnx - ∫(x^4/5)dx
= (1/5)(x^5)lnx - x^5/25 + C
= (x^5/5){ln x - 1/5} + C

Answer 2

yes integrate them by parts. u know the integral of ln x, its found by taking 1 as second and ln x as first function.
when u will integrate x^4 lnx by parts taking ln x as second then u will get x^5 lnx - 1/5 x^5 - 4∫x^4 ln x dx
take - 4∫x^4 ln x dx to L.H.S. and u have the answer
well i dont know where do u get 1/5 x^4, but its anti derivative is

1/4 * 1/5 x^5 = 1/20 x^5

Answer 3

u = ln x
du = 1/x dx

dv = x^4 dx
v = (1/5) x^5

Int[x^4 lnx dx]
= uv - Int[vdu]
= (1/5) x^5 lnx - Int[(1/5) x^5 1/x dx]
= (1/5) x^5 lnx - (1/5) Int[x^4 dx]
= (1/5) x^5 lnx - (1/5) (1/5) x^5 + C
= (1/5) x^5 lnx - (1/25) x^5 + C

Answer 4

integral (antiderivative) of x^n is { [ x^(n+1)] / (n+1 ) }

so integral (antiderivative) of (1/5)x^4 is (1/25)x^5

a solution to your problem is available at the following link

Answer 5

Integrate by parts.

d(uv) = u dv + v du ----> (uv) = Int (u) dv + Int (v) du ---->
(uv) - Int (v) du = Int (u) dv

Let u = ln x. Then du = (1/x) dx
Let dv = x^4 dx. Then v = (x^5)/5.

So Int (u) dv = ln x (x^5)/5 - Int [(x^5)/5] (1/x) dx ----->
Int (u) dv = ln x (x^5)/5 - Int [(x^4)/5] dx ---->
Int (u) dv = ln x (x^5)/5 - {1/5[(x^5)/5]} + c ---->
Int (u) dv = ln x (x^5)/5 - (x^5)/25 + c ---->
Int (u) dv = [(x^5)/5] [ln x - 1/5] + c

So Int [(ln x)(x^4)] dx = [(x^5)/5] [ln x - 1/5] + c

Answer 6

you set dv= x^4 dx and u= lnx

so v = x^5/5 and du = 1/x dx

you have int udv = uv- int vdu

int (x^4(lnx) dx) = (x^5/5 ) lnx - int(( x^5/5) /x )dx

= (x^5/5 ) lnx- int(x^4/5) dx

int (x^4(lnx) dx) =(x^5/5 ) lnx - x^5/25

Correct check my link