Primitive of Power of x by Logarithm of a x

Theorem

$\ds \int x^n \ln a x \rd x = \dfrac {x^{n + 1} } {n + 1} \ln a x - \dfrac {x^{n + 1} } {\paren {n + 2}^2} + C$

for $n \ne -1$.

Proof

With a view to expressing the primitive in the form:

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

let:

\(\ds u\)

\(=\)

\(\ds \ln a x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds \frac 1 x\)

Derivative of $\ln a x$

and let:

\(\ds \frac {\d v} {\d x}\)

\(=\)

\(\ds x^n\)

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

\(\ds \frac {x^{n + 1} } {n + 1}\)

Primitive of Power

Then:

\(\ds \int x^n \ln a x \rd x\)

\(=\)

\(\ds \frac {x^{n + 1} } {n + 1} \ln a x - \int \frac {x^{n + 1} } {n + 1} \paren {\frac 1 x} \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds \frac {x^{n + 1} } {n + 1} \ln a x - \frac 1 {n + 1} \int x^n \rd x\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds \frac {x^{n + 1} } {n + 1} \ln a x - \frac 1 {n + 1} \paren {\frac {x^{n + 1} } {n + 1} } + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac {x^{n + 1} } {n + 1} \paren {\ln a x - \frac 1 {n + 1} } + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\dfrac {\ln a x} x$ for $n = -1$

• Primitive of $x^n \ln x$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $110$.