What is the integral from e^y to y of (yln(x))/x ?

The integral from e^y to y of (yln(x))/x is (y(ln^2(y)-y^2))/2

integral from e^y to y of (yln(x))/x

\int_{e^{y}}^{y} \frac{y ln ( x )}{x} d x

\frac{y ( ln ^{2} ( y ) - y ^{2} )}{2}

Solution steps

\int_{e^{y}}^{y} \frac{y ln ( x )}{x} d x

Take the constant out:

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

= y \cdot \int_{e^{y}}^{y} \frac{ln ( x )}{x} d x

Apply u-substitution

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

Apply the Power Rule

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

Compute the boundaries:

\frac{ln ^{2} ( y )}{2} - \frac{y ^{2}}{2}

= y (\frac{ln ^{2} ( y )}{2} - \frac{y ^{2}}{2})

Simplify

= \frac{y ( ln ^{2} ( y ) - y ^{2} )}{2}

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What is the integral from e^y to y of (yln(x))/x ?
The integral from e^y to y of (yln(x))/x is (y(ln^2(y)-y^2))/2