5.5 The Substitution Rule/30

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{\sin{(\ln{(x))}}}{x}dx }

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 \begin{align} u &=2+x^4 \\[2ex] du &= 4x^3dx \\[2ex] \frac{1}{4}du &= x^3dx \end{align} }

${\begin{aligned}\int x^{3}(2+x^{4})^{5}dx&=\int (x^{3}dx)(2+x^{4})=\int \left({\frac {1}{4}}du\right)(u)={\frac {1}{4}}\int u\,du\\[2ex]&={\frac {1}{4}}\left[{\frac {u^{1+1}}{1+1}}\right]+C={\frac {u^{2}}{8}}+C\\[2ex]&={\frac {(2+x^{4})^{2}}{8}}+C\end{aligned}}$