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Solving integrals/ (4x^3-3x^2+5)

Integral of (4x^3-3x^2+5) dx

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The solution

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  3                     
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 |  /   3      2    \   
 |  \4*x  - 3*x  + 5/ dx
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l
∫l3((4x3−3x2)+5) dx\int\limits_{l}^{3} \left(\left(4 x^{3} - 3 x^{2}\right) + 5\right)\, dx 
Integral(4*x^3 - 3*x^2 + 5, (x, l, 3))
Detail solution

  1. Integrate term-by-term:

    1. Integrate term-by-term:

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

        ∫4x3 dx=4∫x3 dx\int 4 x^{3}\, dx = 4 \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: x4x^{4}

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

        ∫(−3x2) dx=−3∫x2 dx\int \left(- 3 x^{2}\right)\, dx = - 3 \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: −x3- x^{3}

      The result is: x4−x3x^{4} - x^{3}

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

      ∫5 dx=5x\int 5\, dx = 5 x

    The result is: x4−x3+5xx^{4} - x^{3} + 5 x

  2. Now simplify:

    x(x3−x2+5)x \left(x^{3} - x^{2} + 5\right)

  3. Add the constant of integration:

    x(x3−x2+5)+constantx \left(x^{3} - x^{2} + 5\right)+ \mathrm{constant}

The answer is:

x(x3−x2+5)+constantx \left(x^{3} - x^{2} + 5\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
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 | /   3      2    \           4    3      
 | \4*x  - 3*x  + 5/ dx = C + x  - x  + 5*x
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∫((4x3−3x2)+5) dx=C+x4−x3+5x\int \left(\left(4 x^{3} - 3 x^{2}\right) + 5\right)\, dx = C + x^{4} - x^{3} + 5 x
Simplify
The answer [src] 
      3    4      
69 + l  - l  - 5*l
−l4+l3−5l+69- l^{4} + l^{3} - 5 l + 69 
=
= 
      3    4      
69 + l  - l  - 5*l
−l4+l3−5l+69- l^{4} + l^{3} - 5 l + 69 
69 + l^3 - l^4 - 5*l

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

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