Primitive of Square of Logarithm of x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \ln^2 x \rd x = x \ln^2 x - 2 x \ln x + 2 x + C$


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:

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


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \ln x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds x \ln x - x\) Primitive of $\ln x$


Then:

\(\ds \int \ln^2 x \rd x\) \(=\) \(\ds \int \paren {\ln x} \paren {\ln x} \rd x\)
\(\ds \) \(=\) \(\ds \paren {\ln x} \paren {x \ln x - x} - \int \paren {x \ln x - x} \frac 1 x \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds x \ln^2 x - x \ln x - \int \ln x \rd x + \int \rd x + C\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds x \ln^2 x - x \ln x - \paren {x \ln x - x} + \int \rd x + C\) Primitive of $\ln x$
\(\ds \) \(=\) \(\ds x \ln^2 x - x \ln x - \paren {x \ln x - x} + x + C\) Primitive of Constant
\(\ds \) \(=\) \(\ds x \ln^2 x - 2 x \ln x + 2 x + C\) simplifying

$\blacksquare$


Sources