7.1 Integration By Parts/51

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 \text{use Exercise 47 to evaluate} \int(\ln{x})^3dx }

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 \text{Exercise 47:}\qquad x(\ln{x})^n-n\int(\ln{x})^{n-1}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 \int(\ln{x})^3dx= x(\ln{x})^3-3\int(\ln{x})^2dx }

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 u = (\ln{x})^2 \qquad dv = 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 du = 2\frac{1}{x}\ln{x}dx \qquad v = x }

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} \int(\ln{x})^3dx &= x(\ln{x})^3-3\int(\ln{x})^2dx = x(\ln{x})^3-3[x(\ln{x})^2 - 2\int(\ln{x})dx] \\[2ex] &= x(\ln{x})^3-3[x(\ln{x})^2 -2(x\ln{x} - \int1dx) = x(\ln{x})^3-3[x(\ln{x})^2 -2x\ln{x} + 2x] \\[2ex] & =x(\ln{x})^3 - 3x(\ln{x})^2+ 6x\ln{x} - 6x +c \\[2ex] \end{align} }