What is the integral from 2x^2 to 1+x^2 of (x+2y) ?

The integral from 2x^2 to 1+x^2 of (x+2y) is x-x^3+1+2x^2-3x^4

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integral from 2x^2 to 1+x^2 of (x+2y)

\int_{2 x^{2}}^{1 + x^{2}} (x + 2 y) d y

x - x^{3} + 1 + 2 x^{2} - 3 x^{4}

Solution steps

\int_{2 x^{2}}^{1 + x^{2}} x + 2 y 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 x^{2}}^{1 + x^{2}} x d y + \int_{2 x^{2}}^{1 + x^{2}} 2 y d y

\int_{2 x^{2}}^{1 + x^{2}} x d y = x - x^{3}

\int_{2 x^{2}}^{1 + x^{2}} 2 y d y = 1 + 2 x^{2} - 3 x^{4}

= x - x^{3} + 1 + 2 x^{2} - 3 x^{4}

What is the integral from 2x^2 to 1+x^2 of (x+2y) ?
The integral from 2x^2 to 1+x^2 of (x+2y) is x-x^3+1+2x^2-3x^4