Question
Integrate x^-3(x+1)dx
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Answered by
GPT 3.5
To integrate x^-3(x+1)dx, we first expand the expression:
x^-3(x+1) = x^-2 + x^-3
Now, we can integrate each term separately:
∫ x^-2 dx = ∫ 1/x^2 dx = -1/x + C
∫ x^-3 dx = ∫ 1/x^3 dx = -1/(2x^2) + C
Therefore, the integral of x^-3(x+1)dx is equal to:
-1/x + (-1/(2x^2)) + C
which simplifies to:
-1/x - 1/(2x^2) + C
So, ∫ x^-3(x+1)dx = -1/x - 1/(2x^2) + C
x^-3(x+1) = x^-2 + x^-3
Now, we can integrate each term separately:
∫ x^-2 dx = ∫ 1/x^2 dx = -1/x + C
∫ x^-3 dx = ∫ 1/x^3 dx = -1/(2x^2) + C
Therefore, the integral of x^-3(x+1)dx is equal to:
-1/x + (-1/(2x^2)) + C
which simplifies to:
-1/x - 1/(2x^2) + C
So, ∫ x^-3(x+1)dx = -1/x - 1/(2x^2) + C
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