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sin(3x^2)
Solving integrals/ sin(3x^2)

Integral of sin(3x^2) dx

Limits of integration:

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v

The graph:

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Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |     /   2\   
 |  sin\3*x / dx
 |              
/               
0
$$\int\limits_{0}^{1} \sin{\left(3 x^{2} \right)}\, dx$$
Simplify
Detail solution

    FresnelSRule(a=3, b=0, c=0, context=sin(3*x**2), symbol=x)

  1. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
                                    /    ___\
                        ___   ____  |x*\/ 6 |
  /                   \/ 6 *\/ pi *S|-------|
 |                                  |   ____|
 |    /   2\                        \ \/ pi /
 | sin\3*x / dx = C + -----------------------
 |                               6           
/
$${{\sqrt{\pi}\,\left(\left(\sqrt{2}\,\sqrt{3}\,i+\sqrt{2}\,\sqrt{3} \right)\,\mathrm{erf}\left({{\left(\sqrt{2}\,\sqrt{3}\,i+\sqrt{2}\, \sqrt{3}\right)\,x}\over{2}}\right)+\left(\sqrt{2}\,\sqrt{3}\,i- \sqrt{2}\,\sqrt{3}\right)\,\mathrm{erf}\left({{\left(\sqrt{2}\, \sqrt{3}\,i-\sqrt{2}\,\sqrt{3}\right)\,x}\over{2}}\right)+\left( \sqrt{2}\,\sqrt{3}-\sqrt{2}\,\sqrt{3}\,i\right)\,\mathrm{erf}\left( \sqrt{3}\,\sqrt{-i}\,x\right)+\left(\sqrt{2}\,\sqrt{3}\,i+\sqrt{2}\, \sqrt{3}\right)\,\mathrm{erf}\left(\left(-1\right)^{{{1}\over{4}}}\, \sqrt{3}\,x\right)\right)}\over{48}}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
              /  ___ \           
  ___   ____  |\/ 6  |           
\/ 6 *\/ pi *S|------|*Gamma(3/4)
              |  ____|           
              \\/ pi /           
---------------------------------
           8*Gamma(7/4)
$${{\sqrt{\pi}\,\left(\left(\sqrt{2}\,\sqrt{3}\,i+\sqrt{2}\,\sqrt{3} \right)\,\mathrm{erf}\left({{\sqrt{2}\,\sqrt{3}\,i+\sqrt{2}\,\sqrt{3 }}\over{2}}\right)+\left(\sqrt{2}\,\sqrt{3}\,i-\sqrt{2}\,\sqrt{3} \right)\,\mathrm{erf}\left({{\sqrt{2}\,\sqrt{3}\,i-\sqrt{2}\,\sqrt{3 }}\over{2}}\right)+\left(\sqrt{2}\,\sqrt{3}-\sqrt{2}\,\sqrt{3}\,i \right)\,\mathrm{erf}\left(\sqrt{3}\,\sqrt{-i}\right)+\left(\sqrt{2} \,\sqrt{3}\,i+\sqrt{2}\,\sqrt{3}\right)\,\mathrm{erf}\left(\left(-1 \right)^{{{1}\over{4}}}\,\sqrt{3}\right)\right)}\over{48}}$$ 
=
= 
              /  ___ \           
  ___   ____  |\/ 6  |           
\/ 6 *\/ pi *S|------|*Gamma(3/4)
              |  ____|           
              \\/ pi /           
---------------------------------
           8*Gamma(7/4)
$$\frac{\sqrt{6} \sqrt{\pi} S\left(\frac{\sqrt{6}}{\sqrt{\pi}}\right) \Gamma\left(\frac{3}{4}\right)}{8 \Gamma\left(\frac{7}{4}\right)}$$
Simplify
Numerical answer [src] 
0.514976174485537
0.514976174485537
The graph
Integral of sin(3x^2) dx

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

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