What is the integral from 2 to 4 of (xy^2+x^2+y+4) ?

The integral from 2 to 4 of (xy^2+x^2+y+4) is 2x^2+(56x)/3+14

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integral from 2 to 4 of (xy^2+x^2+y+4)

\int_{2}^{4} (x y^{2} + x^{2} + y + 4) d y

2 x^{2} + \frac{56 x}{3} + 14

Solution steps

\int_{2}^{4} x y^{2} + x^{2} + y + 4 d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{2}^{4} 4 d y + \int_{2}^{4} x^{2} d y + \int_{2}^{4} y d y + \int_{2}^{4} x y^{2} d y

\int_{2}^{4} 4 d y = 8

\int_{2}^{4} x^{2} d y = 2 x^{2}

\int_{2}^{4} y d y = 6

\int_{2}^{4} x y^{2} d y = \frac{56 x}{3}

= 8 + 2 x^{2} + 6 + \frac{56 x}{3}

Simplify

= 2 x^{2} + \frac{56 x}{3} + 14

What is the integral from 2 to 4 of (xy^2+x^2+y+4) ?
The integral from 2 to 4 of (xy^2+x^2+y+4) is 2x^2+(56x)/3+14