खण्डशः समाकलन

कलनमे खण्डशः समाकलन (integration by parts) एक प्रमेय हे जे दु फलनके गुणनफलके समाकलके निम्नलिखित प्रकारसे व्यक्त करहे -

$\int u(x)v'(x)\,dx=u(x)v(x)-\int v(x)\,u'(x)dx$

उपरोक्तके छोट रूपमे निम्नलिखितो ढङ्गसे लिखल जाहे -

$\int u\,dv=uv-\int v\,du.\!$

उदाहरण

• $\int xe^{x}dx=xe^{x}-\int e^{x}dx=(x-1)e^{x}\,$ + C

• $\int _{1}^{2}x\ln(x)dx=\left[{\frac {x^{2}}{2}}\ln(x)\right]_{1}^{2}-{\frac {1}{2}}\int _{1}^{2}xdx\,$

${\begin{aligned}\int e^{x}\cdot \left(2-x^{2}\right)\,\mathrm {d} x&=e^{x}\cdot \left(2-x^{2}\right)-\int e^{x}\cdot (-2x)\,\mathrm {d} x\\&=e^{x}\cdot \left(2-x^{2}\right)+e^{x}\cdot 2x-\int 2\cdot e^{x}\,\mathrm {d} x\\&=e^{x}\cdot \left(2-x^{2}\right)+e^{x}\cdot 2x-2\cdot e^{x}+C\\&=e^{x}\cdot \left(2-x^{2}+2x-2\right)+C\\&=e^{x}\cdot \left(2x-x^{2}\right)+C\,.\end{aligned}}$

• $\int \sin ^{5}x\;\cos ^{2}xdx.$

$\sin ^{5}x\;\cos ^{2}x=(\sin ^{2}x)^{2}\;\cos ^{2}x\;\sin x=(1-\cos ^{2}x)^{2}\;\cos ^{2}x\;\sin x$

${\begin{matrix}\int \sin ^{5}x\cos ^{2}xdx=\int \sin ^{4}x\cos ^{2}x\sin x\ dx=\\\int (1-\cos ^{2}x)^{2}\cos ^{2}x\sin x\ dx=\\\int (1-u^{2})^{2}\;u^{2}\;(-du)=-\int (u^{2}-2u^{4}+u^{6})du=\\-({\frac {u^{3}}{3}}-{\frac {2u^{5}}{5}}+{\frac {u^{7}}{7}})+C=\\-{\frac {1}{3}}\cos ^{3}x+{\frac {2}{5}}\cos ^{5}x-{\frac {1}{7}}\cos ^{7}x+C\end{matrix}}$

इहो देखी

• समाकलन