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  • Identical expressions

  • two pi*x^ two *sqrt(one +4x^2)
  • 2 Pi multiply by x squared multiply by square root of (1 plus 4x squared )
  • two Pi multiply by x to the power of two multiply by square root of (one plus 4x squared )
  • 2pi*x^2*√(1+4x^2)
  • 2pi*x2*sqrt(1+4x2)
  • 2pi*x2*sqrt1+4x2
  • 2pi*x²*sqrt(1+4x²)
  • 2pi*x to the power of 2*sqrt(1+4x to the power of 2)
  • 2pix^2sqrt(1+4x^2)
  • 2pix2sqrt(1+4x2)
  • 2pix2sqrt1+4x2
  • 2pix^2sqrt1+4x^2
  • 2pi*x^2*sqrt(1+4x^2)dx

  • Similar expressions

  • 2pi*x^2*sqrt(1-4x^2)
Solving integrals/ 2pi*x^2*sqrt(1+4x^2)

Integral of 2pi*x^2*sqrt(1+4x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                         
  /                         
 |                          
 |             __________   
 |        2   /        2    
 |  2*pi*x *\/  1 + 4*x   dx
 |                          
/                           
0
$$\int\limits_{0}^{1} 2 \pi x^{2} \sqrt{4 x^{2} + 1}\, dx$$ 
Integral(((2*pi)*x^2)*sqrt(1 + 4*x^2), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                                                                                 
 |                                                                                                  
 |            __________                                   5                                   3    
 |       2   /        2           pi*asinh(2*x)      2*pi*x             pi*x             3*pi*x     
 | 2*pi*x *\/  1 + 4*x   dx = C - ------------- + ------------- + ---------------- + ---------------
 |                                      32           __________         __________        __________
/                                                   /        2         /        2        /        2 
                                                  \/  1 + 4*x     16*\/  1 + 4*x     4*\/  1 + 4*x
$$\int 2 \pi x^{2} \sqrt{4 x^{2} + 1}\, dx = C + \frac{2 \pi x^{5}}{\sqrt{4 x^{2} + 1}} + \frac{3 \pi x^{3}}{4 \sqrt{4 x^{2} + 1}} + \frac{\pi x}{16 \sqrt{4 x^{2} + 1}} - \frac{\pi \operatorname{asinh}{\left(2 x \right)}}{32}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
     /                 ___\
     |  asinh(2)   9*\/ 5 |
2*pi*|- -------- + -------|
     \     64         32  /
$$2 \pi \left(- \frac{\operatorname{asinh}{\left(2 \right)}}{64} + \frac{9 \sqrt{5}}{32}\right)$$ 
=
= 
     /                 ___\
     |  asinh(2)   9*\/ 5 |
2*pi*|- -------- + -------|
     \     64         32  /
$$2 \pi \left(- \frac{\operatorname{asinh}{\left(2 \right)}}{64} + \frac{9 \sqrt{5}}{32}\right)$$ 
2*pi*(-asinh(2)/64 + 9*sqrt(5)/32)
Numerical answer [src] 
3.8097297048578
3.8097297048578

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

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