UV আকারের (Integration by parts)
∫x2sinxdx=?\int x^{2} \sin x d x = ?∫x2sinxdx=?
−x2cosx+2xsinx+2cosx+c-x^{2} \cos x+2 x \sin x+2 \cos x+c−x2cosx+2xsinx+2cosx+c
x2cosx+2xsinx+2cosx+cx^{2} \cos x+2 x \sin x+2 \cos x+cx2cosx+2xsinx+2cosx+c
−x2cosx−2xsinx+2cosx+c-x^{2} \cos x-2 x \sin x+2 \cos x+c−x2cosx−2xsinx+2cosx+c
−x2cosx+2xsinx−2cosx+c-x^{2} \cos x+2 x \sin x-2 \cos x+c−x2cosx+2xsinx−2cosx+c
Solve:
∫x2sinxdx=x2∫sinxdx−∫{ddx(x2)∫sinxdx}dx=x2(−cosx)−∫2x(−cosx)dx=−x2cosx+2[x∫cosx−∫−{ddx(x)∫cosxdx}}dx]=−x2cosx+2[xsinx−∫1⋅sinxdx]=−x2cosx+2[xsinx−(−cosx)]+c=−x2cosx+2xsinx+2cosx+c \begin{array}{l} \int x^{2} \sin x d x \\ =x^{2} \int \sin x d x-\int\left\{\frac{d}{d x}\left(x^{2}\right) \int \sin x d x\right\} d x \\ =x^{2}(-\cos x)-\int 2 x(-\cos x) d x \\ =-x^{2} \cos x+2\left[x \int \cos x-\right. \\ \left.\left.\int-\left\{\frac{d}{d x}(x) \int \cos x d x\right\}\right\} d x\right] \\ =-x^{2} \cos x+2\left[x \sin x-\int 1 \cdot \sin x d x\right] \\ =-x^{2} \cos x+2[x \sin x-(-\cos x)]+c \\ =-x^{2} \cos x+2 x \sin x+2 \cos x+c \end{array} ∫x2sinxdx=x2∫sinxdx−∫{dxd(x2)∫sinxdx}dx=x2(−cosx)−∫2x(−cosx)dx=−x2cosx+2[x∫cosx−∫−{dxd(x)∫cosxdx}}dx]=−x2cosx+2[xsinx−∫1⋅sinxdx]=−x2cosx+2[xsinx−(−cosx)]+c=−x2cosx+2xsinx+2cosx+c
∫xcos−1x2dx=? \int x \cos ^{-1} x^{2} d x = ?∫xcos−1x2dx=?
∫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)=?
The sequence S0,S1,S2S_0,S_1,S_2S0,S1,S2.... forms a G.P with common ratio
