What is the integral of 2xy+x^2-1 ?

The integral of 2xy+x^2-1 is -y+x^2y+xy^2+C

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integral of 2xy+x^2-1

\int 2 x y + x^{2} - 1 d y

- y + x^{2} y + x y^{2} + C

Solution steps

\int 2 x y + x^{2} - 1 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 d y + \int x^{2} d y + \int 2 x y d y

\int 1 d y = y

\int x^{2} d y = x^{2} y

\int 2 x y d y = x y^{2}

= - y + x^{2} y + x y^{2}

Add a constant to the solution

= - y + x^{2} y + x y^{2} + C

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