5.5 The Substitution Rule/21

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int \frac{\cos{(\sqrt{t})}}{\sqrt{t}}dt \ = \ \int (\sqrt{u})du \\[2ex] }

${\begin{aligned}&\int {\frac {\left(\ln(x)\right)^{2}}{x}}dx\ =\ \int u^{2}du\\[2ex]&=\ {\frac {u^{2+1}}{2+1}}du\ =\ {\frac {1}{3}}u^{3}+C\\[2ex]&u=\ln(x)\\&du={\frac {1}{x}}dx\\&=\ {\frac {1}{3}}(\ln(x))^{3}+C\end{aligned}}$