What is the integral of x^4 / (x-1)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the integral of x^4/(x - 1)

Int[x^4/(x - 1) dx]

=> Int[(x^4 - 1 + 1)/(x - 1) dx]

=> Int[(x^4 - 1)/(x - 1) + 1/(x - 1) dx]

=> Int[(x^2 - 1)(x^2 + 1)/(x - 1) + 1/(x - 1) dx]

=> Int[(x - 1)(x + 1)(x^2 + 1)/(x - 1) + 1/(x - 1) dx]

=> Int[(x + 1)(x^2 + 1) + 1/(x - 1) dx]

=> Int[x^3 + x^2  + x + 1) + 1/(x - 1) dx]

=> Int[x^3 + x^2  + x + 1) dx] + Int[1/(x - 1) dx]

=> x^4/4 + x^3/3 + x^2/2 + x + log(x - 1)

The required integral is x^4/4 + x^3/3 + x^2/2 + x + log(x - 1) + C

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