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Solving integrals/ 3*x*sin(y)+1

Integral of 3*x*sin(y)+1 dx

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The solution

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∫01(3xsin⁡(y)+1) dx\int\limits_{0}^{1} \left(3 x \sin{\left(y \right)} + 1\right)\, dx 
Integral((3*x)*sin(y) + 1, (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:

      ∫3xsin⁡(y) dx=sin⁡(y)∫3x dx\int 3 x \sin{\left(y \right)}\, dx = \sin{\left(y \right)} \int 3 x\, dx

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

        ∫3x dx=3∫x dx\int 3 x\, dx = 3 \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: 3x22\frac{3 x^{2}}{2}

      So, the result is: 3x2sin⁡(y)2\frac{3 x^{2} \sin{\left(y \right)}}{2}

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    The result is: 3x2sin⁡(y)2+x\frac{3 x^{2} \sin{\left(y \right)}}{2} + x

  2. Now simplify:

    x(3xsin⁡(y)+2)2\frac{x \left(3 x \sin{\left(y \right)} + 2\right)}{2}

  3. Add the constant of integration:

    x(3xsin⁡(y)+2)2+constant\frac{x \left(3 x \sin{\left(y \right)} + 2\right)}{2}+ \mathrm{constant}

The answer is:

x(3xsin⁡(y)+2)2+constant\frac{x \left(3 x \sin{\left(y \right)} + 2\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 2       
 |                               3*x *sin(y)
 | (3*x*sin(y) + 1) dx = C + x + -----------
 |                                    2     
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∫(3xsin⁡(y)+1) dx=C+3x2sin⁡(y)2+x\int \left(3 x \sin{\left(y \right)} + 1\right)\, dx = C + \frac{3 x^{2} \sin{\left(y \right)}}{2} + x
Simplify
The answer [src] 
    3*sin(y)
1 + --------
       2
3sin⁡(y)2+1\frac{3 \sin{\left(y \right)}}{2} + 1 
=
= 
    3*sin(y)
1 + --------
       2
3sin⁡(y)2+1\frac{3 \sin{\left(y \right)}}{2} + 1 
1 + 3*sin(y)/2

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

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