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

The integral from x^2 to 2x of (x^2+y^2) is 2x^3-x^4+(8x^3-x^6)/3

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

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

2 x^{3} - x^{4} + \frac{8 x ^{3} - x ^{6}}{3}

Solution steps

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

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

\int_{x^{2}}^{2 x} y^{2} d y = \frac{8 x ^{3} - x ^{6}}{3}

= 2 x^{3} - x^{4} + \frac{8 x ^{3} - x ^{6}}{3}

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What is the integral from x^2 to 2x of (x^2+y^2) ?
The integral from x^2 to 2x of (x^2+y^2) is 2x^3-x^4+(8x^3-x^6)/3