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Solving integrals/ (x-3)*(x-5)

Integral of (x-3)*(x-5) dx

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The solution

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∫01(x−5)(x−3) dx\int\limits_{0}^{1} \left(x - 5\right) \left(x - 3\right)\, dx 
Integral((x - 3)*(x - 5), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    (x−5)(x−3)=x2−8x+15\left(x - 5\right) \left(x - 3\right) = x^{2} - 8 x + 15

  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:

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

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

      ∫(−8x) dx=−8∫x dx\int \left(- 8 x\right)\, dx = - 8 \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: −4x2- 4 x^{2}

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

      ∫15 dx=15x\int 15\, dx = 15 x

    The result is: x33−4x2+15x\frac{x^{3}}{3} - 4 x^{2} + 15 x

  3. Now simplify:

    x(x2−12x+45)3\frac{x \left(x^{2} - 12 x + 45\right)}{3}

  4. Add the constant of integration:

    x(x2−12x+45)3+constant\frac{x \left(x^{2} - 12 x + 45\right)}{3}+ \mathrm{constant}

The answer is:

x(x2−12x+45)3+constant\frac{x \left(x^{2} - 12 x + 45\right)}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | (x - 3)*(x - 5) dx = C - 4*x  + 15*x + --
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∫(x−5)(x−3) dx=C+x33−4x2+15x\int \left(x - 5\right) \left(x - 3\right)\, dx = C + \frac{x^{3}}{3} - 4 x^{2} + 15 x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90020
Plot the graph f(x)
The answer [src] 
34/3
343\frac{34}{3} 
=
= 
34/3
343\frac{34}{3} 
34/3
Numerical answer [src] 
11.3333333333333
11.3333333333333

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

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