UV আকারের (Integration by parts)
∫e2xcosexdx=?\int e^{2 x} \cos e^{x} d x = ?∫e2xcosexdx=?
exsinex+cosex+ce^{x} \sin e^{x}+\cos e^{x}+cexsinex+cosex+c
−exsinex+cosex+c-e^{x} \sin e^{x}+\cos e^{x}+c−exsinex+cosex+c
exsinex−cosex+ce^{x} \sin e^{x}-\cos e^{x}+cexsinex−cosex+c
−exsinex−cosex+c-e^{x} \sin e^{x}-\cos e^{x}+c−exsinex−cosex+c
Solve: ধরি, I=∫e2xcosexdx \mathrm{I}=\int e^{2 x} \cos e^{x} d x I=∫e2xcosexdx এবং ex=z e^{x}=z ex=z.তাহলে exdx=dz e^{x} d x=d z exdx=dz,এবং
I=∫excosex(exdx)=∫zcoszdz=z∫coszdz−∫{ddz(z)∫coszdz}dz=zsinz−∫1⋅sinzdz=zsinz−(−cosz)+c=exsinex+cosex+c \begin{array}{l} \mathrm{I}=\int e^{x} \cos e^{x}\left(e^{x} d x\right)=\int z \cos z d z \\ =z \int \cos z d z-\int\left\{\frac{d}{d z}(z) \int \cos z d z\right\} d z \\ =z \sin z-\int 1 \cdot \sin z d z \\ =z \sin z-(-\cos z)+c \\ =e^{x} \sin e^{x}+\cos e^{x}+c \end{array} I=∫excosex(exdx)=∫zcoszdz=z∫coszdz−∫{dzd(z)∫coszdz}dz=zsinz−∫1⋅sinzdz=zsinz−(−cosz)+c=exsinex+cosex+c
∫xcos−1x2dx=? \int x \cos ^{-1} x^{2} d x = ?∫xcos−1x2dx=?
∫x3sinxdx=?\int x^{3} \sin x d x = ?∫x3sinxdx=?
∫ln(1+x)1+xdx equals\displaystyle\int {\dfrac{{\ln \left( {1 + {x}} \right)}}{{1 + {x}}}} dx\,equals∫1+xln(1+x)dxequals
∫x3exdx=f(x)+c \int x^{3} e^{x} dx = f{\left ( x \right )} + c ∫x3exdx=f(x)+c হয় তবে f(x)=?
