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Identical expressions
xysin two y+xsiny^2- four
xy sinus of 2y plus x sinus of y squared minus 4
xy sinus of two y plus x sinus of y squared minus four
xysin2y+xsiny2-4
xysin2y+xsiny²-4
xysin2y+xsiny to the power of 2-4
xysin2y+xsiny^2-4dx
Similar expressions
xysin2y-xsiny^2-4
xysin2y+xsiny^2+4

Integral of xysin2y+xsiny^2-4 dx

The solution

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  1                                  
  /                                  
 |                                   
 |  /                    2       \   
 |  \x*y*sin(2*y) + x*sin (y) - 4/ dx
 |                                   
/                                    
0

$$\int\limits_{0}^{1} \left(x y \sin{\left(2 y \right)} + x \sin^{2}{\left(y \right)} - 4\right)\, dx$$

Integral(x*y*sin(2*y) + x*sin(y)^2 - 1*4, (x, 0, 1))

Detail solution

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

$\frac{x \left(2 x y \sin{\left(2 y \right)} - x \cos{\left(2 y \right)} + x - 16\right)}{4}$
Add the constant of integration:

$\frac{x \left(2 x y \sin{\left(2 y \right)} - x \cos{\left(2 y \right)} + x - 16\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{x \left(2 x y \sin{\left(2 y \right)} - x \cos{\left(2 y \right)} + x - 16\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                        
 |                                                2    2         2         
 | /                    2       \                x *sin (y)   y*x *sin(2*y)
 | \x*y*sin(2*y) + x*sin (y) - 4/ dx = C - 4*x + ---------- + -------------
 |                                                   2              2      
/

$$\int \left(x y \sin{\left(2 y \right)} + x \sin^{2}{\left(y \right)} - 4\right)\, dx = C + \frac{x^{2} y \sin{\left(2 y \right)}}{2} + \frac{x^{2} \sin^{2}{\left(y \right)}}{2} - 4 x$$

Simplify

The answer [src]

        2                
     sin (y)   y*sin(2*y)
-4 + ------- + ----------
        2          2

$$\frac{y \sin{\left(2 y \right)}}{2} + \frac{\sin^{2}{\left(y \right)}}{2} - 4$$

        2                
     sin (y)   y*sin(2*y)
-4 + ------- + ----------
        2          2

$$\frac{y \sin{\left(2 y \right)}}{2} + \frac{\sin^{2}{\left(y \right)}}{2} - 4$$

Упростить

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

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Integral of xysin2y+xsiny^2-4 dx

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Piecewise:

The solution