posted by .

Could someone work this question out so I understand it.

Find the indefinite intregral

(lnx)^7 / x dx

Use C as the arbitrary constant.

  • Math -

    The best method here is integration by parts. Most of the times when you see products of functions AND the product of one with the others derivative gives a simple to integrate function.

    Do you see that [ln(x)]' = 1/x. And when you multiply it with a power of x you get a power of x?

    The standard form of the integration by parts is:

    integral_{f'(x)*g(x)}dx = f(x)*g(x) - integral_{f(x)*g'(x)}dx

    The idea is to pick f and g so it'll all be simpler. We already picked out g(x) = ln(x)

    Now we need f'(x). This can only be x^7. But what functions derivative is x^7? We have to to the opposite of derivation ---> integration!
    f(x) = integral_{f'(x)}dx = integral_{x^7}dx = (x^8)/8.
    Just check! [(x^8)/8]' = [(1/8)*(x^8)]' = (1/8)*8*(x^7) = x^7

    integral_{f'(x)*g(x)}dx = integral_{(x^7)*ln(x)}dx = integral_{ [(x^8)/8]' * ln(x) }dx =

    = [(x^8)/8] * ln(x) - integral_{ [(x^8)/8] * ln'(x) }dx =

    = [(x^8)/8] * ln(x) - integral_{ [(1/8)*(x^8)] * 1/x }dx =

    = [(x^8)/8] * ln(x) - (1/8)* integral_{ [x^(8-1)] }dx =

    = [(x^8)/8] * ln(x) - (1/8)* integral_{ [x^7] }dx =

    = [(x^8)/8] * ln(x) - (1/8)*(x^8)/8 + real_constant =

    = [(x^8)/8] * ln(x) - (x^8)/64 + real_constant = [(x^8)/8] * [ln(x) - (1/8)] + real_constant
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = [(x^8)/64] * [ 8*ln(x) - 1 ] + real_constant

  • Math -

    Sometimes there are different ways to solve a problem, here is another:

    ∫(lnx)^7 / x dx
    We note that (lnx)' = 1/x
    and substitute y=ln(x), dy=dx/x
    ∫(lnx)^7 / x dx
    = ∫ y^7 dy
    = y^8/8 + C
    = ln(x)^8/8 + C

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Math- Still Don't Understand!

    I do not understand how to work this problem out. I have a series of other questions that are similar, and it would really help me out if someone could help me understand how to work this problem. At the first tri-city meeting, there …
  2. Math

    Find the coordinates of the turning point and determine wether it is minimum or maximum. y=xlnx-2x The answer in the book says the co-ordinates are (e,-e), the closest I have come is (1/lnx,-1/lnx) which works if 1/lnx=e, but I don't …
  3. math

    Could someone answer this question. im stuck on it. Id appreciate it. Use substitution to find the indefinite integral. çp(p+3)^5 dp Use C as the arbitrary constant.
  4. Math

    Could someone answer this question so I understand it. Thanks Find the indefinite integral: çx/ã3x^2+4 dx Use C as the arbitrary constant.
  5. Math

    Could someone answer this question so I understand it. Thanks Find the indefinite integral. x/ã3x^2+4 dx = Simplify the answer. Use C as the arbitrary constant.
  6. Math

    Evaluate the following indefinite integral. Use C as the arbitrary constant. ç(12 /ãx+12ãx)dx Only the x's are under the root symbols. I got 6x^1/2 +8x^3/2+C as my answer. Is that correct?
  7. Math

    I cant figure this question out. I could use some help. Solve for x lnx-lnx^2+ln5 = 0 Simplify the answer
  8. L'Hopital's rule

    Find lim x->1+ of [(1/(x-1))-(1/lnx)]. Here is my work... =(lnx-(x-1)) / ((x-1)(lnx)) =(lnx-1) / (lnx+ (x+1)/x) This becomes(1/x) / ((1/x)+(1/x^2)) which becomes 1/ (1/x^2) This equals 1/2. I understand the answer has to be -1/2, …
  9. Calculus 1

    Find the derivative of y with respect to x. y=(x^6/6)(lnx)-(x^6/36) So far this is what I've gotten: y=(x^6/6)(lnx)-(x^6/36) y=(1/6)x^6(lnx)-(1/36)x^6 y'=(1/6)x^5(1/x)+lnx(x^5)-(1/6)x^5 What do I do now?
  10. calculus

    sorry to ask a second question so soon, but i'm just not getting this one. if f(x)= 3x lnx, then f'(x)=?

More Similar Questions