Integral of ₂∫⁴(4x³-2x²)dx dx

The solution

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$$\int\limits_{0}^{1} 16 \left(4 x^{3} - 2 x^{2}\right)\, dx$$

Integral(16*(4*x^3 - 2*x^2), (x, 0, 1))

Detail solution

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

$\int 16 \left(4 x^{3} - 2 x^{2}\right)\, dx = 16 \int \left(4 x^{3} - 2 x^{2}\right)\, dx$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 x^{3}\, dx = 4 \int x^{3}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
    So, the result is: $x^{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $- \frac{2 x^{3}}{3}$
  The result is: $x^{4} - \frac{2 x^{3}}{3}$
So, the result is: $16 x^{4} - \frac{32 x^{3}}{3}$
Now simplify:

$x^{3} \left(16 x - \frac{32}{3}\right)$
Add the constant of integration:

$x^{3} \left(16 x - \frac{32}{3}\right)+ \mathrm{constant}$

The answer is:

$x^{3} \left(16 x - \frac{32}{3}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

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 |    /   3      2\              4   32*x 
 | 16*\4*x  - 2*x / dx = C + 16*x  - -----
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$$\int 16 \left(4 x^{3} - 2 x^{2}\right)\, dx = C + 16 x^{4} - \frac{32 x^{3}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

16/3

$$\frac{16}{3}$$

16/3

$$\frac{16}{3}$$

16/3

Numerical answer [src]

5.33333333333333

5.33333333333333

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

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Integral of ₂∫⁴(4x³-2x²)dx dx

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The solution