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(2x+7)^8

Integral of (2x+7)^8 dx

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01(2x+7)8dx\int\limits_{0}^{1} \left(2 x + 7\right)^{8}\, dx
Integral((2*x + 7)^8, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2x+7u = 2 x + 7.

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

      u82du\int \frac{u^{8}}{2}\, du

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

        u8du=u8du2\int u^{8}\, du = \frac{\int u^{8}\, du}{2}

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

          u8du=u99\int u^{8}\, du = \frac{u^{9}}{9}

        So, the result is: u918\frac{u^{9}}{18}

      Now substitute uu back in:

      (2x+7)918\frac{\left(2 x + 7\right)^{9}}{18}

    Method #2

    1. Rewrite the integrand:

      (2x+7)8=256x8+7168x7+87808x6+614656x5+2689120x4+7529536x3+13176688x2+13176688x+5764801\left(2 x + 7\right)^{8} = 256 x^{8} + 7168 x^{7} + 87808 x^{6} + 614656 x^{5} + 2689120 x^{4} + 7529536 x^{3} + 13176688 x^{2} + 13176688 x + 5764801

    2. Integrate term-by-term:

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

        256x8dx=256x8dx\int 256 x^{8}\, dx = 256 \int x^{8}\, dx

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

          x8dx=x99\int x^{8}\, dx = \frac{x^{9}}{9}

        So, the result is: 256x99\frac{256 x^{9}}{9}

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

        7168x7dx=7168x7dx\int 7168 x^{7}\, dx = 7168 \int x^{7}\, dx

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

          x7dx=x88\int x^{7}\, dx = \frac{x^{8}}{8}

        So, the result is: 896x8896 x^{8}

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

        87808x6dx=87808x6dx\int 87808 x^{6}\, dx = 87808 \int x^{6}\, dx

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

          x6dx=x77\int x^{6}\, dx = \frac{x^{7}}{7}

        So, the result is: 12544x712544 x^{7}

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

        614656x5dx=614656x5dx\int 614656 x^{5}\, dx = 614656 \int x^{5}\, dx

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

          x5dx=x66\int x^{5}\, dx = \frac{x^{6}}{6}

        So, the result is: 307328x63\frac{307328 x^{6}}{3}

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

        2689120x4dx=2689120x4dx\int 2689120 x^{4}\, dx = 2689120 \int x^{4}\, dx

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

          x4dx=x55\int x^{4}\, dx = \frac{x^{5}}{5}

        So, the result is: 537824x5537824 x^{5}

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

        7529536x3dx=7529536x3dx\int 7529536 x^{3}\, dx = 7529536 \int x^{3}\, dx

        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}

        So, the result is: 1882384x41882384 x^{4}

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

        13176688x2dx=13176688x2dx\int 13176688 x^{2}\, dx = 13176688 \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: 13176688x33\frac{13176688 x^{3}}{3}

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

        13176688xdx=13176688xdx\int 13176688 x\, dx = 13176688 \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: 6588344x26588344 x^{2}

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

        5764801dx=5764801x\int 5764801\, dx = 5764801 x

      The result is: 256x99+896x8+12544x7+307328x63+537824x5+1882384x4+13176688x33+6588344x2+5764801x\frac{256 x^{9}}{9} + 896 x^{8} + 12544 x^{7} + \frac{307328 x^{6}}{3} + 537824 x^{5} + 1882384 x^{4} + \frac{13176688 x^{3}}{3} + 6588344 x^{2} + 5764801 x

  2. Now simplify:

    (2x+7)918\frac{\left(2 x + 7\right)^{9}}{18}

  3. Add the constant of integration:

    (2x+7)918+constant\frac{\left(2 x + 7\right)^{9}}{18}+ \mathrm{constant}


The answer is:

(2x+7)918+constant\frac{\left(2 x + 7\right)^{9}}{18}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                              
 |                              9
 |          8          (2*x + 7) 
 | (2*x + 7)  dx = C + ----------
 |                         18    
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(2x+7)8dx=C+(2x+7)918\int \left(2 x + 7\right)^{8}\, dx = C + \frac{\left(2 x + 7\right)^{9}}{18}
The graph
0.001.000.100.200.300.400.500.600.700.800.90050000000
The answer [src]
173533441/9
1735334419\frac{173533441}{9}
=
=
173533441/9
1735334419\frac{173533441}{9}
173533441/9
Numerical answer [src]
19281493.4444444
19281493.4444444
The graph
Integral of (2x+7)^8 dx

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