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(x^2-4)/x
Solving integrals/ (x^2-4)/x

Integral of (x^2-4)/x dx

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∫01x2−4x dx\int\limits_{0}^{1} \frac{x^{2} - 4}{x}\, dx 
Integral((x^2 - 4)/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2u = x^{2}.

      Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

      ∫u−42u du\int \frac{u - 4}{2 u}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫u−4u du=∫u−4u du2\int \frac{u - 4}{u}\, du = \frac{\int \frac{u - 4}{u}\, du}{2}

        1. Rewrite the integrand:

          u−4u=1−4u\frac{u - 4}{u} = 1 - \frac{4}{u}

        2. Integrate term-by-term:

          1. The integral of a constant is the constant times the variable of integration:

            ∫1 du=u\int 1\, du = u

          1. The integral of a constant times a function is the constant times the integral of the function:

            ∫(−4u) du=−4∫1u du\int \left(- \frac{4}{u}\right)\, du = - 4 \int \frac{1}{u}\, du

            1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

            So, the result is: −4log⁡(u)- 4 \log{\left(u \right)}

          The result is: u−4log⁡(u)u - 4 \log{\left(u \right)}

        So, the result is: u2−2log⁡(u)\frac{u}{2} - 2 \log{\left(u \right)}

      Now substitute uu back in:

      x22−2log⁡(x2)\frac{x^{2}}{2} - 2 \log{\left(x^{2} \right)}

    Method #2

    1. Rewrite the integrand:

      x2−4x=x−4x\frac{x^{2} - 4}{x} = x - \frac{4}{x}

    2. Integrate term-by-term:

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−4x) dx=−4∫1x dx\int \left(- \frac{4}{x}\right)\, dx = - 4 \int \frac{1}{x}\, dx

        1. The integral of 1x\frac{1}{x} is log⁡(x)\log{\left(x \right)}.

        So, the result is: −4log⁡(x)- 4 \log{\left(x \right)}

      The result is: x22−4log⁡(x)\frac{x^{2}}{2} - 4 \log{\left(x \right)}

  2. Add the constant of integration:

    x22−2log⁡(x2)+constant\frac{x^{2}}{2} - 2 \log{\left(x^{2} \right)}+ \mathrm{constant}

The answer is:

x22−2log⁡(x2)+constant\frac{x^{2}}{2} - 2 \log{\left(x^{2} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                              
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∫x2−4x dx=C+x22−2log⁡(x2)\int \frac{x^{2} - 4}{x}\, dx = C + \frac{x^{2}}{2} - 2 \log{\left(x^{2} \right)}
Simplify
The graph
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The answer [src] 
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Numerical answer [src] 
-175.861784535972
-175.861784535972
The graph
Integral of (x^2-4)/x dx

    Use the examples entering the upper and lower limits of integration.

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