Mister Exam

Other calculators


(2x-3)^7

Integral of (2x-3)^7 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1              
  /              
 |               
 |           7   
 |  (2*x - 3)  dx
 |               
/                
0                
01(2x3)7dx\int\limits_{0}^{1} \left(2 x - 3\right)^{7}\, dx
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2x3u = 2 x - 3.

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

      u74du\int \frac{u^{7}}{4}\, du

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

        u72du=u7du2\int \frac{u^{7}}{2}\, du = \frac{\int u^{7}\, du}{2}

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

          u7du=u88\int u^{7}\, du = \frac{u^{8}}{8}

        So, the result is: u816\frac{u^{8}}{16}

      Now substitute uu back in:

      (2x3)816\frac{\left(2 x - 3\right)^{8}}{16}

    Method #2

    1. Rewrite the integrand:

      (2x3)7=128x71344x6+6048x515120x4+22680x320412x2+10206x2187\left(2 x - 3\right)^{7} = 128 x^{7} - 1344 x^{6} + 6048 x^{5} - 15120 x^{4} + 22680 x^{3} - 20412 x^{2} + 10206 x - 2187

    2. Integrate term-by-term:

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

        128x7dx=128x7dx\int 128 x^{7}\, dx = 128 \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: 16x816 x^{8}

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

        (1344x6)dx=1344x6dx\int \left(- 1344 x^{6}\right)\, dx = - 1344 \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: 192x7- 192 x^{7}

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

        6048x5dx=6048x5dx\int 6048 x^{5}\, dx = 6048 \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: 1008x61008 x^{6}

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

        (15120x4)dx=15120x4dx\int \left(- 15120 x^{4}\right)\, dx = - 15120 \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: 3024x5- 3024 x^{5}

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

        22680x3dx=22680x3dx\int 22680 x^{3}\, dx = 22680 \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: 5670x45670 x^{4}

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

        (20412x2)dx=20412x2dx\int \left(- 20412 x^{2}\right)\, dx = - 20412 \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: 6804x3- 6804 x^{3}

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

        10206xdx=10206xdx\int 10206 x\, dx = 10206 \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: 5103x25103 x^{2}

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

        (2187)dx=2187x\int \left(-2187\right)\, dx = - 2187 x

      The result is: 16x8192x7+1008x63024x5+5670x46804x3+5103x22187x16 x^{8} - 192 x^{7} + 1008 x^{6} - 3024 x^{5} + 5670 x^{4} - 6804 x^{3} + 5103 x^{2} - 2187 x

  2. Now simplify:

    (2x3)816\frac{\left(2 x - 3\right)^{8}}{16}

  3. Add the constant of integration:

    (2x3)816+constant\frac{\left(2 x - 3\right)^{8}}{16}+ \mathrm{constant}


The answer is:

(2x3)816+constant\frac{\left(2 x - 3\right)^{8}}{16}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                              
 |                              8
 |          7          (2*x - 3) 
 | (2*x - 3)  dx = C + ----------
 |                         16    
/                                
(2x3)816{{\left(2\,x-3\right)^8}\over{16}}
The graph
0.001.000.100.200.300.400.500.600.700.800.90-40002000
The answer [src]
-410
410-410
=
=
-410
410-410
Numerical answer [src]
-410.0
-410.0
The graph
Integral of (2x-3)^7 dx

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