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Question
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Solution
Let I = `int x^7/(x + 1)*dx`
= `int ((x^7 + 1) - 1)/(x + 1)*dx`
= `int ((x + 1)(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) - 1)/(x + 1)*dx`
= `int [x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 - (1)/(x + 1)]*dx`
= `int x^6 dx - intx^5 dx + intx^4 dx - intx^3 dx + intx^2 dx - intx dx + int1 dx - int (1)/(x + 1) dx`
= `x^7/(7) - x^6/(6) + x^5/(5) - x^4/(4) + x^3/(3) - x^2/(2) + x - log|x + 1| + c`.
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