What is the integral from 1 to x of 24x^2y^2-48xy ?

The integral from 1 to x of 24x^2y^2-48xy is 8x^2(x^3-1)-24x(x^2-1)

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integral from 1 to x of 24x^2y^2-48xy

\int_{1}^{x} 24 x^{2} y^{2} - 48 x y d y

8 x^{2} (x^{3} - 1) - 24 x (x^{2} - 1)

Solution steps

\int_{1}^{x} 24 x^{2} y^{2} - 48 x 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_{1}^{x} 24 x^{2} y^{2} d y - \int_{1}^{x} 48 x y d y

\int_{1}^{x} 24 x^{2} y^{2} d y = 8 x^{2} (x^{3} - 1)

\int_{1}^{x} 48 x y d y = 24 x (x^{2} - 1)

= 8 x^{2} (x^{3} - 1) - 24 x (x^{2} - 1)

\int_{2}^{3} \frac{x ^{2}}{x ^{2} - 1} d x

\int_{0}^{4} x^{2} - 9 d x

\int_{0}^{1} 4 x^{2} + 1 x d x

\int_{4}^{5} 16 - x^{2} d x

\int_{0}^{10} \frac{1}{1 + x ^{2}} d x

What is the integral from 1 to x of 24x^2y^2-48xy ?
The integral from 1 to x of 24x^2y^2-48xy is 8x^2(x^3-1)-24x(x^2-1)