Integral of (4x³+x)√4x²+1dx dx

The solution

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  1                             
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 |  /                  2    \   
 |  |/   3    \   _____     |   
 |  \\4*x  + x/*\/ 4*x   + 1/ dx
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0

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

Integral((4*x^3 + x)*(sqrt(4*x))^2 + 1, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Rewrite the integrand:
  
  $\left(4 x^{3} + x\right) \left(\sqrt{4 x}\right)^{2} = 16 x^{4} + 4 x^{2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 16 x^{4}\, dx = 16 \int x^{4}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
    So, the result is: $\frac{16 x^{5}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$
    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: $\frac{4 x^{3}}{3}$
  The result is: $\frac{16 x^{5}}{5} + \frac{4 x^{3}}{3}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $\frac{16 x^{5}}{5} + \frac{4 x^{3}}{3} + x$
Now simplify:

$\frac{x \left(48 x^{4} + 20 x^{2} + 15\right)}{15}$
Add the constant of integration:

$\frac{x \left(48 x^{4} + 20 x^{2} + 15\right)}{15}+ \mathrm{constant}$

The answer is:

$\frac{x \left(48 x^{4} + 20 x^{2} + 15\right)}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
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 | /                  2    \                 3       5
 | |/   3    \   _____     |              4*x    16*x 
 | \\4*x  + x/*\/ 4*x   + 1/ dx = C + x + ---- + -----
 |                                         3       5  
/

$$\int \left(\left(4 x^{3} + x\right) \left(\sqrt{4 x}\right)^{2} + 1\right)\, dx = C + \frac{16 x^{5}}{5} + \frac{4 x^{3}}{3} + x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

83
--
15

$$\frac{83}{15}$$

83
--
15

$$\frac{83}{15}$$

83/15

Numerical answer [src]

5.53333333333333

5.53333333333333

The graph

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

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Integral of (4x³+x)√4x²+1dx dx

Limits of integration:

The graph:

Piecewise:

The solution