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(x+1)^5
Solving integrals/ (x+1)^5

Integral of (x+1)^5 dx

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The solution

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∫01(x+1)5 dx\int\limits_{0}^{1} \left(x + 1\right)^{5}\, dx 
Integral((x + 1)^5, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+1u = x + 1.

      Then let du=dxdu = dx and substitute dudu:

      ∫u5 du\int u^{5}\, du

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

        ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

      Now substitute uu back in:

      (x+1)66\frac{\left(x + 1\right)^{6}}{6}

    Method #2

    1. Rewrite the integrand:

      (x+1)5=x5+5x4+10x3+10x2+5x+1\left(x + 1\right)^{5} = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1

    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:

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

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

        ∫5x4 dx=5∫x4 dx\int 5 x^{4}\, dx = 5 \int x^{4}\, dx

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

          ∫x4 dx=x55\int x^{4}\, dx = \frac{x^{5}}{5}

        So, the result is: x5x^{5}

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

        ∫10x3 dx=10∫x3 dx\int 10 x^{3}\, dx = 10 \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: 5x42\frac{5 x^{4}}{2}

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

        ∫10x2 dx=10∫x2 dx\int 10 x^{2}\, dx = 10 \int x^{2}\, dx

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

          ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

        So, the result is: 10x33\frac{10 x^{3}}{3}

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

        ∫5x dx=5∫x dx\int 5 x\, dx = 5 \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: 5x22\frac{5 x^{2}}{2}

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

        ∫1 dx=x\int 1\, dx = x

      The result is: x66+x5+5x42+10x33+5x22+x\frac{x^{6}}{6} + x^{5} + \frac{5 x^{4}}{2} + \frac{10 x^{3}}{3} + \frac{5 x^{2}}{2} + x

  2. Now simplify:

    (x+1)66\frac{\left(x + 1\right)^{6}}{6}

  3. Add the constant of integration:

    (x+1)66+constant\frac{\left(x + 1\right)^{6}}{6}+ \mathrm{constant}

The answer is:

(x+1)66+constant\frac{\left(x + 1\right)^{6}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                          
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 |        5          (x + 1) 
 | (x + 1)  dx = C + --------
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∫(x+1)5 dx=C+(x+1)66\int \left(x + 1\right)^{5}\, dx = C + \frac{\left(x + 1\right)^{6}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90050
Plot the graph f(x)
The answer [src] 
21/2
212\frac{21}{2} 
=
= 
21/2
212\frac{21}{2} 
21/2
Numerical answer [src] 
10.5
10.5
The graph
Integral of (x+1)^5 dx

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

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