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Integral of (x+3)^3 dx

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The solution

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10(x+3)3dx\int\limits_{1}^{0} \left(x + 3\right)^{3}\, dx
Integral((x + 3)^3, (x, 1, 0))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+3u = 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:

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

    Method #2

    1. Rewrite the integrand:

      (x+3)3=x3+9x2+27x+27\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:

        9x2dx=9x2dx\int 9 x^{2}\, 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: 3x33 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:

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

      The result is: x44+3x3+27x22+27x\frac{x^{4}}{4} + 3 x^{3} + \frac{27 x^{2}}{2} + 27 x

  2. Now simplify:

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

  3. Add the constant of integration:

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


The answer is:

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

The answer (Indefinite) [src]
  /                          
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 |        3          (x + 3) 
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(x+3)3dx=C+(x+3)44\int \left(x + 3\right)^{3}\, dx = C + \frac{\left(x + 3\right)^{4}}{4}
The graph
0.001.000.100.200.300.400.500.600.700.800.900100
The answer [src]
-175/4
1754- \frac{175}{4}
=
=
-175/4
1754- \frac{175}{4}
-175/4
Numerical answer [src]
-43.75
-43.75

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