What is the integral of e^{2x}ln(x) ?

The integral of e^{2x}ln(x) is 1/2 e^{2x}ln(x)-1/2 \Ei(2x)+C

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integral of e^{2x}ln(x)

\int e^{2 x} ln (x) d x

\frac{1}{2} e^{2 x} ln (x) - \frac{1}{2} Ei (2 x) + C

Solution steps

\int e^{2 x} ln (x) d x

Apply Integration By Parts

: \frac{1}{2} e^{2 x} ln (x) - \int \frac{e ^{2 x}}{2 x} d x

= \frac{1}{2} e^{2 x} ln (x) - \int \frac{e ^{2 x}}{2 x} d x

\int \frac{e ^{2 x}}{2 x} d x = \frac{1}{2} Ei (2 x)

= \frac{1}{2} e^{2 x} ln (x) - \frac{1}{2} Ei (2 x)

Add a constant to the solution

= \frac{1}{2} e^{2 x} ln (x) - \frac{1}{2} Ei (2 x) + C

\int (\frac{4}{x} + sec^{2} (x)) d x

x y^{^{'}} + y = e^{x}, y (1) = 8

cos^{2} (5 π x)

\frac{5 ln ( 4 x )}{2 - 3 x}

\int \frac{4}{5} - \frac{8}{x} d x

What is the integral of e^{2x}ln(x) ?
The integral of e^{2x}ln(x) is 1/2 e^{2x}ln(x)-1/2 \Ei(2x)+C