Primitive of Exponential of a x by Logarithm of x

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$\ds \int e^{a x} \ln x \rd x = \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C$


With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$


\(\ds u\) \(=\) \(\ds \ln x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac 1 x\) Derivative of Natural Logarithm

and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds e^{a x}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {e^{a x} } a\) Primitive of $e^{a x}$


\(\ds \int e^{a x} \ln x \rd x\) \(=\) \(\ds \ln x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \frac 1 x \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {e^{a x} \ln x} a - \frac 1 a \int \frac {e^{a x} } x \rd x + C\) Primitive of Constant Multiple of Function


Also see