Using integration by parts, show that for any positive integer n, ∫ x n e n = x n e x − n ∫ x n − 1 e x d xUse the above formula to determine ∫ x 4 e x d x

Question

Using integration by parts, show that for any positive integer n,   ∫      x    n        e    n    =      x    n        e    x    −  n  ∫      x          n      −      1            e    x    d  xUse the above formula to determine   ∫      x    4        e    x    d  x

Bradley Sherman · Accepted Answer

Step 1  I  =  ∫      x    n        e    x    d  xWe know integration by past formula.  ∫  u  v  d  x  =  u  ∫  v  d  x  −  ∫            d      y              d      x        ⋅  ∫  v  d  x  ⋅  dHere consider   u  =      x    n   and   v  =      e    x    I  =  ∫      x    n        e    x    d  x  =      x    n        e    x    −  ∫  n      x          n      −      1        ,      e    x    d  x  ∫      x    n        e    x    d  x  =      x    n        e    x    −  n  ∫      x          n      −      1            e    x    d  xStep 2  ∫      x    4        e    x    d  x  =      x    4        e    a    −  4  ∫      x    3        e    x    d  x  =      x    4        e    x    −  4  (      x    3        e    x    −  8  ∫      x    2        e    x    d  x  )  =      x    4        e    x    −  4      x    3        e    x    +  12  (      x    2        e    x    −  2  ∫  x      e    x    d  x  )  =      x    4        e    x    −  4      x    3        e    x    +  12      x    2        e    x    −  24  (  x      e    x    −  ∫      e    x    d  x  )  =      x    4        e    x    −  4      x    3        e    x    +  12      x    2        e    x    −  24  x      e    x    +  24      e    x    +  c  ∫      x    4        e    x    d  x  =      e    x    (      x    4    −  4      x    3    +  12      x    2    −  24  x  +  24  )  +  c

Using integration by parts, show that for any positive integer n, int x^n e^xdx=x^n e^x-n int x^{n-1}e^xdx

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