What is the integral from 0 to ln(x) of xy ?

The integral from 0 to ln(x) of xy is (xln^2(x))/2

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integral from 0 to ln(x) of xy

\int_{0}^{l n (x)} x y d y

\frac{x ln ^{2} ( x )}{2}

Solution steps

\int_{0}^{l n (x)} x y d y

Take the constant out:

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

= x \cdot \int_{0}^{l n (x)} y d y

Apply the Power Rule

: [\frac{y ^{2}}{2}]_{0}^{l n (x)}

= x [\frac{y ^{2}}{2}]_{0}^{l n (x)}

Compute the boundaries:

\frac{ln ^{2} ( x )}{2}

= x \frac{ln ^{2} ( x )}{2}

Simplify

= \frac{x ln ^{2} ( x )}{2}

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What is the integral from 0 to ln(x) of xy ?
The integral from 0 to ln(x) of xy is (xln^2(x))/2