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Solving integrals/ 3x(y^2+4)

Integral of 3x(y^2+4) dy

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The solution

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∫153x(y2+4) dy\int\limits_{1}^{5} 3 x \left(y^{2} + 4\right)\, dy 
Integral((3*x)*(y^2 + 4), (y, 1, 5))
Detail solution

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

    ∫3x(y2+4) dy=3x∫(y2+4) dy\int 3 x \left(y^{2} + 4\right)\, dy = 3 x \int \left(y^{2} + 4\right)\, dy

    1. Integrate term-by-term:

      1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫y2 dy=y33\int y^{2}\, dy = \frac{y^{3}}{3}

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

        ∫4 dy=4y\int 4\, dy = 4 y

      The result is: y33+4y\frac{y^{3}}{3} + 4 y

    So, the result is: 3x(y33+4y)3 x \left(\frac{y^{3}}{3} + 4 y\right)

  2. Now simplify:

    xy(y2+12)x y \left(y^{2} + 12\right)

  3. Add the constant of integration:

    xy(y2+12)+constantx y \left(y^{2} + 12\right)+ \mathrm{constant}

The answer is:

xy(y2+12)+constantx y \left(y^{2} + 12\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
 |                           /       3\
 |     / 2    \              |      y |
 | 3*x*\y  + 4/ dy = C + 3*x*|4*y + --|
 |                           \      3 /
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∫3x(y2+4) dy=C+3x(y33+4y)\int 3 x \left(y^{2} + 4\right)\, dy = C + 3 x \left(\frac{y^{3}}{3} + 4 y\right)
Simplify
The answer [src] 
172*x
172x172 x 
=
= 
172*x
172x172 x 
172*x

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

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