Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §7.1 Integration by Parts, #56 · Calculus · Techniques of integration

… is the …th derivative of …, and … is the …th…

These problems need more than one integration.

After $n$ integrations by parts, $\int u (d v / d x) d x$ becomes

$uv - u^{(1)} v_{(1)} + u^{(2)} v_{(2)} - \dots + (- 1)^{n} \int u^{(n)} v_{(n - 1)} d x$ .

$u^{(n)}$ is the $n$ th derivative of $u$ , and $v_{(n)}$ is the $n$ th integral of $v$ . Integrate the last term by parts to stretch this formula to $n + 1$ integrations.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §7.1 Integration by Parts, #56

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