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Solving integrals/ x*sina+y*sinx

Integral of x*sina+y*sinx dx

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The solution

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 |  (x*sin(a) + y*sin(x)) dx
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∫01(xsin⁡(a)+ysin⁡(x)) dx\int\limits_{0}^{1} \left(x \sin{\left(a \right)} + y \sin{\left(x \right)}\right)\, dx 
Integral(x*sin(a) + y*sin(x), (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫xsin⁡(a) dx=sin⁡(a)∫x dx\int x \sin{\left(a \right)}\, dx = \sin{\left(a \right)} \int x\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: x2sin⁡(a)2\frac{x^{2} \sin{\left(a \right)}}{2}

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫ysin⁡(x) dx=y∫sin⁡(x) dx\int y \sin{\left(x \right)}\, dx = y \int \sin{\left(x \right)}\, dx

      1. The integral of sine is negative cosine:

        ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

      So, the result is: −ycos⁡(x)- y \cos{\left(x \right)}

    The result is: x2sin⁡(a)2−ycos⁡(x)\frac{x^{2} \sin{\left(a \right)}}{2} - y \cos{\left(x \right)}

  2. Add the constant of integration:

    x2sin⁡(a)2−ycos⁡(x)+constant\frac{x^{2} \sin{\left(a \right)}}{2} - y \cos{\left(x \right)}+ \mathrm{constant}

The answer is:

x2sin⁡(a)2−ycos⁡(x)+constant\frac{x^{2} \sin{\left(a \right)}}{2} - y \cos{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                2                  
 |                                x *sin(a)           
 | (x*sin(a) + y*sin(x)) dx = C + --------- - y*cos(x)
 |                                    2               
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∫(xsin⁡(a)+ysin⁡(x)) dx=C+x2sin⁡(a)2−ycos⁡(x)\int \left(x \sin{\left(a \right)} + y \sin{\left(x \right)}\right)\, dx = C + \frac{x^{2} \sin{\left(a \right)}}{2} - y \cos{\left(x \right)}
Simplify
The answer [src] 
    sin(a)           
y + ------ - y*cos(1)
      2
−ycos⁡(1)+y+sin⁡(a)2- y \cos{\left(1 \right)} + y + \frac{\sin{\left(a \right)}}{2} 
=
= 
    sin(a)           
y + ------ - y*cos(1)
      2
−ycos⁡(1)+y+sin⁡(a)2- y \cos{\left(1 \right)} + y + \frac{\sin{\left(a \right)}}{2} 
y + sin(a)/2 - y*cos(1)

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

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