7.1 Integration By Parts/24

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 u\,dv= u\cdot v -\int v\, du }

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_{0}^{\pi}\underbrace{x^3\cos(x)}_{ \begin{aligned} u&=x^3 \quad \quad dv=\cos(x) \\ dv&=3x^2 \quad \quad v=\sin(x) \end{aligned}} \,dx =x^3\sin(x)-\int_{0}^{\pi} \underbrace{3x^2\sin(x)}_{ \begin{aligned} u&=3x^2 \quad \quad dv=\sin(x) \\ du&=6x \quad \quad v=-\cos(x) \end{aligned}} \,dx= x^3\sin(x)-\bigg[3x^2-\cos(x)-\int_{0}^{\pi}-6x\cos(x)\,dx\bigg]\\ =&x^3\sin(x)-3x^2\cos(x)-\int_{0}^{\pi}\underbrace{6x\cos(x)}_{ \begin{aligned} u&=6x \quad \quad dv=cos(x) \\ du&=6 \quad \quad v=sin(x) \end{aligned}} =x^3\sin(x)+3x^2\cos(x)-\bigg[6x\sin(x)-\int_{0}^{\pi} 6\sin(x)\,dx\bigg]= x^3sin(x)+3x^2\cos(x)-6x\sin(x)-6\cos(x)|_{0}^{\pi} \end{align} }