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x*(x^2-1)^3
Solving integrals/ x*(x^2-1)^3

Integral of x*(x^2-1)^3 dx

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∫01x(x2−1)3 dx\int\limits_{0}^{1} x \left(x^{2} - 1\right)^{3}\, dx 
Integral(x*(x^2 - 1*1)^3, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2−1u = x^{2} - 1.

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

      ∫u34 du\int \frac{u^{3}}{4}\, du

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

        ∫u32 du=∫u3 du2\int \frac{u^{3}}{2}\, du = \frac{\int u^{3}\, du}{2}

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

        So, the result is: u48\frac{u^{4}}{8}

      Now substitute uu back in:

      (x2−1)48\frac{\left(x^{2} - 1\right)^{4}}{8}

    Method #2

    1. Rewrite the integrand:

      x(x2−1)3=x7−3x5+3x3−xx \left(x^{2} - 1\right)^{3} = x^{7} - 3 x^{5} + 3 x^{3} - 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:

        ∫x7 dx=x88\int x^{7}\, dx = \frac{x^{8}}{8}

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

        ∫(−3x5) dx=−3∫x5 dx\int \left(- 3 x^{5}\right)\, dx = - 3 \int x^{5}\, dx

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

          ∫x5 dx=x66\int x^{5}\, dx = \frac{x^{6}}{6}

        So, the result is: −x62- \frac{x^{6}}{2}

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

        ∫3x3 dx=3∫x3 dx\int 3 x^{3}\, dx = 3 \int x^{3}\, dx

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

          ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

        So, the result is: 3x44\frac{3 x^{4}}{4}

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

        ∫(−x) dx=−∫x dx\int \left(- x\right)\, dx = - \int x\, dx

        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}

        So, the result is: −x22- \frac{x^{2}}{2}

      The result is: x88−x62+3x44−x22\frac{x^{8}}{8} - \frac{x^{6}}{2} + \frac{3 x^{4}}{4} - \frac{x^{2}}{2}

  2. Now simplify:

    (x2−1)48\frac{\left(x^{2} - 1\right)^{4}}{8}

  3. Add the constant of integration:

    (x2−1)48+constant\frac{\left(x^{2} - 1\right)^{4}}{8}+ \mathrm{constant}

The answer is:

(x2−1)48+constant\frac{\left(x^{2} - 1\right)^{4}}{8}+ \mathrm{constant}

The answer (Indefinite) [src] 
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(x2−1)48{{\left(x^2-1\right)^4}\over{8}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.25-0.25
Plot the graph f(x)
The answer [src] 
-1/8
−18-{{1}\over{8}} 
=
= 
-1/8
−18- \frac{1}{8}
Упростить
Numerical answer [src] 
-0.125
-0.125
The graph
Integral of x*(x^2-1)^3 dx

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

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