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cos(x^2-y^2)

Integral of cos(x^2+y^2) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |     / 2    2\   
 |  cos\x  + y / dx
 |                 
/                  
0

$$\int\limits_{0}^{1} \cos{\left(x^{2} + y^{2} \right)}\, dx$$

Integral(cos(x^2 + y^2), (x, 0, 1))

Detail solution

FresnelCRule(a=1, b=0, c=y**2, context=cos(x**2 + y**2), symbol=x)

Add the constant of integration:

$\frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(y^{2} \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y^{2} \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(y^{2} \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y^{2} \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                      /         /    ___\    /    ___\        \
                           ___   ____ |   / 2\  |x*\/ 2 |    |x*\/ 2 |    / 2\|
  /                      \/ 2 *\/ pi *|cos\y /*C|-------| - S|-------|*sin\y /|
 |                                    |         |   ____|    |   ____|        |
 |    / 2    2\                       \         \ \/ pi /    \ \/ pi /        /
 | cos\x  + y / dx = C + ------------------------------------------------------
 |                                                 2                           
/

$$\int \cos{\left(x^{2} + y^{2} \right)}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(y^{2} \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(y^{2} \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2}$$

Simplify

The answer [src]

             /         /  ___ \    /  ___ \        \
  ___   ____ |   / 2\  |\/ 2  |    |\/ 2  |    / 2\|
\/ 2 *\/ pi *|cos\y /*C|------| - S|------|*sin\y /|
             |         |  ____|    |  ____|        |
             \         \\/ pi /    \\/ pi /        /
----------------------------------------------------
                         2

$$\frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(y^{2} \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) + \cos{\left(y^{2} \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2}$$

             /         /  ___ \    /  ___ \        \
  ___   ____ |   / 2\  |\/ 2  |    |\/ 2  |    / 2\|
\/ 2 *\/ pi *|cos\y /*C|------| - S|------|*sin\y /|
             |         |  ____|    |  ____|        |
             \         \\/ pi /    \\/ pi /        /
----------------------------------------------------
                         2

$$\frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(y^{2} \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) + \cos{\left(y^{2} \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2}$$

sqrt(2)*sqrt(pi)*(cos(y^2)*fresnelc(sqrt(2)/sqrt(pi)) - fresnels(sqrt(2)/sqrt(pi))*sin(y^2))/2

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

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Integral of cos(x^2+y^2) dx

Limits of integration:

The graph:

Piecewise:

The solution