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

The integral from x to x^2+1 of 2xy is x(x^4+x^2+1)

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integral from x to x^2+1 of 2xy

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

x (x^{4} + x^{2} + 1)

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply the Power Rule

= 2 x [\frac{y ^{2}}{2}]_{x}^{1 + x^{2}}

Compute the boundaries:

\frac{( 1 + x ^{2} ) ^{2}}{2} - \frac{x ^{2}}{2}

= 2 x (\frac{( 1 + x ^{2} ) ^{2}}{2} - \frac{x ^{2}}{2})

Simplify

= x (x^{4} + x^{2} + 1)

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