Question

proof of integration by parts​

Answer 1

Step-by-step explanation:

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

If

${\displaystyle u=u(x)}{\displaystyle u=u(x)} and {\displaystyle du=u'(x)\,dx}{\displaystyle du=u'(x)\,dx} while {\displaystyle v=v(x)}{\displaystyle v=v(x)} and {\displaystyle dv=v'(x)dx}{\displaystyle dv=v'(x)dx},$

then the integration by parts formula states that

{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}

More compactly,

{\displaystyle \int u\,dv\ =\ uv-\int v\,du.}{\displaystyle \int u\,dv\ =\ uv-\int v\,du.}

Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.[1][2] More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.

Answer 2

Step-by-step explanation:

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

If {\displaystyle u=u(x)}{\displaystyle u=u(x)} and {\displaystyle du=u'(x)\,dx}{\displaystyle du=u'(x)\,dx} while {\displaystyle v=v(x)}{\displaystyle v=v(x)} and {\displaystyle dv=v'(x)dx}{\displaystyle dv=v'(x)dx}, then the integration by parts formula states that

{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}

More compactly,

{\displaystyle \int u\,dv\ =\ uv-\int v\,du.}{\displaystyle \int u\,dv\ =\ uv-\int v\,du.}

Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.[1][2] More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.