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Kash
Lv 4
Kash asked in Science & MathematicsMathematics · 1 decade ago

What is the integral of 1/x^n?

I know the integral of 1/x is. According to my calculus teacher, the integral of 1/x is ln(x).

My question is, how does the answer differ if the x is risen to a power? What is the integral of 1/x^2 or 1/x^5?

2 Answers

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  • Anonymous
    1 decade ago
    Favorite Answer

    You can think of 1/(x^n) as just x^(-n). So the integral would just be:

    (1 / (-n+1)) x^(-n+1) + C

    For example, the integral of 1 / x^5 is the integral of x^(-5), or

    -(1/4) x^(-4) + C

    1/x is just a special case where the power rule doesn't apply, because you can't say the integral of x^-1 is 0*x^(0).

  • Como
    Lv 7
    1 decade ago

    I = ∫ x^(- n) dx

    I = x^(- n + 1) / (- n + 1) + C

    I = x^(1 - n) / (1 - n) + C

    Also

    ∫ 1 / (x ² ) dx = ∫ x^(- 2) dx = x^(-1) / (- 1) + C = - 1 / x + C

    and

    ∫ 1/ x^5 dx = ∫ x^(- 5) dx = x^(- 4) / (- 4) + C = - 1 / (4 x^4) + C

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