Quick Way To Derive The Integration By Parts Formula

Firstly, you need to know what the product rule is:

If\quad y=u\cdot v,\quad \frac { dy }{ dx } =u\frac { dv }{ dx } +v\frac { du }{ dx } .

Then…

\frac { dy }{ dx } =u\frac { dv }{ dx } +v\frac { du }{ dx } \\ \\ u\frac { dv }{ dx } =\frac { dy }{ dx } -v\frac { du }{ dx }

Now integrate each term with respect to x:

\int { u\frac { dv }{ dx } } dx=\int { \frac { dy }{ dx } } dx-\int { v\frac { du }{ dx } } dx

Leaving:

\int { u\frac { dv }{ dx } } dx=y-\int { v\frac { du }{ dx } } dx\\ \\ As\quad y=u\cdot v,\\ \\ \int { u\frac { dv }{ dx } } dx=uv-\int { v\frac { du }{ dx } } dx

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