कैलकुलस में खंडश: समाकलन (integration by parts) एक प्रमेय है जो दो फलनों के गुणनफल के समाकल को निम्नलिखित प्रकार से व्यक्त करता है-
उपरोक्त को छोटे रूप में निम्नलिखित ढंग से भी लिखा जाता है:
∫ e x ⋅ ( 2 − x 2 ) d x = e x ⋅ ( 2 − x 2 ) − ∫ e x ⋅ ( − 2 x ) d x = e x ⋅ ( 2 − x 2 ) + e x ⋅ 2 x − ∫ 2 ⋅ e x d x = e x ⋅ ( 2 − x 2 ) + e x ⋅ 2 x − 2 ⋅ e x + C = e x ⋅ ( 2 − x 2 + 2 x − 2 ) + C = e x ⋅ ( 2 x − x 2 ) + C . {\displaystyle {\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}}}
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 {\displaystyle \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}
∫ sin 5 x cos 2 x d x = ∫ sin 4 x cos 2 x sin x d x = ∫ ( 1 − cos 2 x ) 2 cos 2 x sin x d x = ∫ ( 1 − u 2 ) 2 u 2 ( − d u ) = − ∫ ( u 2 − 2 u 4 + u 6 ) d u = − ( u 3 3 − 2 u 5 5 + u 7 7 ) + C = − 1 3 cos 3 x + 2 5 cos 5 x − 1 7 cos 7 x + C {\displaystyle {\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}}}
+ C![{\displaystyle \int _{1}^{2}x\ln(x)dx=\left[{\frac {x^{2}}{2}}\ln(x)\right]_{1}^{2}-{\frac {1}{2}}\int _{1}^{2}xdx\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cf7b0da913c23197ce0715542fa8787df63a3b)
