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(x-3)^3

Integral of (x-3)^3 dx

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The solution

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14(x3)3dx\int\limits_{1}^{4} \left(x - 3\right)^{3}\, dx
Integral((x - 3)^3, (x, 1, 4))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x3u = x - 3.

      Then let du=dxdu = dx and substitute dudu:

      u3du\int u^{3}\, du

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

        u3du=u44\int u^{3}\, du = \frac{u^{4}}{4}

      Now substitute uu back in:

      (x3)44\frac{\left(x - 3\right)^{4}}{4}

    Method #2

    1. Rewrite the integrand:

      (x3)3=x39x2+27x27\left(x - 3\right)^{3} = x^{3} - 9 x^{2} + 27 x - 27

    2. Integrate term-by-term:

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

        x3dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

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

        (9x2)dx=9x2dx\int \left(- 9 x^{2}\right)\, dx = - 9 \int x^{2}\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

          x2dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

        So, the result is: 3x3- 3 x^{3}

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

        27xdx=27xdx\int 27 x\, dx = 27 \int x\, dx

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

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

        So, the result is: 27x22\frac{27 x^{2}}{2}

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

        (27)dx=27x\int \left(-27\right)\, dx = - 27 x

      The result is: x443x3+27x2227x\frac{x^{4}}{4} - 3 x^{3} + \frac{27 x^{2}}{2} - 27 x

  2. Now simplify:

    (x3)44\frac{\left(x - 3\right)^{4}}{4}

  3. Add the constant of integration:

    (x3)44+constant\frac{\left(x - 3\right)^{4}}{4}+ \mathrm{constant}


The answer is:

(x3)44+constant\frac{\left(x - 3\right)^{4}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                          
 |                          4
 |        3          (x - 3) 
 | (x - 3)  dx = C + --------
 |                      4    
/                            
(x3)3dx=C+(x3)44\int \left(x - 3\right)^{3}\, dx = C + \frac{\left(x - 3\right)^{4}}{4}
The graph
1.004.001.251.501.752.002.252.502.753.003.253.503.75-2525
The answer [src]
-15/4
154- \frac{15}{4}
=
=
-15/4
154- \frac{15}{4}
-15/4
Numerical answer [src]
-3.75
-3.75
The graph
Integral of (x-3)^3 dx

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