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Integral of (5x-3/2)^9 dx

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01(5x32)9dx\int\limits_{0}^{1} \left(5 x - \frac{3}{2}\right)^{9}\, dx
Integral((5*x - 3/2)^9, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=5x32u = 5 x - \frac{3}{2}.

      Then let du=5dxdu = 5 dx and substitute du5\frac{du}{5}:

      u95du\int \frac{u^{9}}{5}\, du

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

        u9du=u9du5\int u^{9}\, du = \frac{\int u^{9}\, du}{5}

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

          u9du=u1010\int u^{9}\, du = \frac{u^{10}}{10}

        So, the result is: u1050\frac{u^{10}}{50}

      Now substitute uu back in:

      (5x32)1050\frac{\left(5 x - \frac{3}{2}\right)^{10}}{50}

    Method #2

    1. Rewrite the integrand:

      (5x32)9=1953125x910546875x82+6328125x78859375x62+15946875x589568125x416+1913625x316492075x232+295245x25619683512\left(5 x - \frac{3}{2}\right)^{9} = 1953125 x^{9} - \frac{10546875 x^{8}}{2} + 6328125 x^{7} - \frac{8859375 x^{6}}{2} + \frac{15946875 x^{5}}{8} - \frac{9568125 x^{4}}{16} + \frac{1913625 x^{3}}{16} - \frac{492075 x^{2}}{32} + \frac{295245 x}{256} - \frac{19683}{512}

    2. Integrate term-by-term:

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

        1953125x9dx=1953125x9dx\int 1953125 x^{9}\, dx = 1953125 \int x^{9}\, dx

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

          x9dx=x1010\int x^{9}\, dx = \frac{x^{10}}{10}

        So, the result is: 390625x102\frac{390625 x^{10}}{2}

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

        (10546875x82)dx=10546875x8dx2\int \left(- \frac{10546875 x^{8}}{2}\right)\, dx = - \frac{10546875 \int x^{8}\, dx}{2}

        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: 1171875x92- \frac{1171875 x^{9}}{2}

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

        6328125x7dx=6328125x7dx\int 6328125 x^{7}\, dx = 6328125 \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: 6328125x88\frac{6328125 x^{8}}{8}

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

        (8859375x62)dx=8859375x6dx2\int \left(- \frac{8859375 x^{6}}{2}\right)\, dx = - \frac{8859375 \int x^{6}\, dx}{2}

        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: 1265625x72- \frac{1265625 x^{7}}{2}

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

        15946875x58dx=15946875x5dx8\int \frac{15946875 x^{5}}{8}\, dx = \frac{15946875 \int x^{5}\, dx}{8}

        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: 5315625x616\frac{5315625 x^{6}}{16}

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

        (9568125x416)dx=9568125x4dx16\int \left(- \frac{9568125 x^{4}}{16}\right)\, dx = - \frac{9568125 \int x^{4}\, dx}{16}

        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: 1913625x516- \frac{1913625 x^{5}}{16}

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

        1913625x316dx=1913625x3dx16\int \frac{1913625 x^{3}}{16}\, dx = \frac{1913625 \int x^{3}\, dx}{16}

        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: 1913625x464\frac{1913625 x^{4}}{64}

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

        (492075x232)dx=492075x2dx32\int \left(- \frac{492075 x^{2}}{32}\right)\, dx = - \frac{492075 \int x^{2}\, dx}{32}

        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: 164025x332- \frac{164025 x^{3}}{32}

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

        295245x256dx=295245xdx256\int \frac{295245 x}{256}\, dx = \frac{295245 \int x\, dx}{256}

        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: 295245x2512\frac{295245 x^{2}}{512}

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

        (19683512)dx=19683x512\int \left(- \frac{19683}{512}\right)\, dx = - \frac{19683 x}{512}

      The result is: 390625x1021171875x92+6328125x881265625x72+5315625x6161913625x516+1913625x464164025x332+295245x251219683x512\frac{390625 x^{10}}{2} - \frac{1171875 x^{9}}{2} + \frac{6328125 x^{8}}{8} - \frac{1265625 x^{7}}{2} + \frac{5315625 x^{6}}{16} - \frac{1913625 x^{5}}{16} + \frac{1913625 x^{4}}{64} - \frac{164025 x^{3}}{32} + \frac{295245 x^{2}}{512} - \frac{19683 x}{512}

  2. Now simplify:

    (10x3)1051200\frac{\left(10 x - 3\right)^{10}}{51200}

  3. Add the constant of integration:

    (10x3)1051200+constant\frac{\left(10 x - 3\right)^{10}}{51200}+ \mathrm{constant}


The answer is:

(10x3)1051200+constant\frac{\left(10 x - 3\right)^{10}}{51200}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                   
 |                                  10
 |            9          (5*x - 3/2)  
 | (5*x - 3/2)  dx = C + -------------
 |                             50     
/                                     
(5x32)9dx=C+(5x32)1050\int \left(5 x - \frac{3}{2}\right)^{9}\, dx = C + \frac{\left(5 x - \frac{3}{2}\right)^{10}}{50}
The graph
0.001.000.100.200.300.400.500.600.700.800.90-100000100000
The answer [src]
1412081
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  256  
1412081256\frac{1412081}{256}
=
=
1412081
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  256  
1412081256\frac{1412081}{256}
1412081/256
Numerical answer [src]
5515.94140625
5515.94140625

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