UV আকারের (Integration by parts)
f(x)=sinx+cosx, g(x)=lnx
∫x2{g(x)}2dx \int x^{2} \left \lbrace g{\left ( x \right )} \right \rbrace^{2} dx ∫x2{g(x)}2dx এর মান নিচের কোনটি ?
e3x(x23−2x3+227) e^{3 x} \left ( \frac{x^{2}}{3} - \frac{2 x}{3} + \frac{2}{27} \right ) e3x(3x2−32x+272)
x3{(lnx)2−2lnx+2}+c x^{3} \left \lbrace \left ( \ln{x} \right )^{2} - 2 \ln{x} + 2 \right \rbrace + c x3{(lnx)2−2lnx+2}+c
x3{(lnx)23−2lnx9−227}+c x^{3} \left \lbrace \frac{\left ( \ln{x} \right )^{2}}{3} - \frac{2 \ln{x}}{9} - \frac{2}{27} \right \rbrace + c x3{3(lnx)2−92lnx−272}+c
∫x2(lnx)2dx=(lnx)2∫x2dx−∫{ddx(lnx)2∫x2⋅dx}dx=x3(lnx)23−23∫lnx2dx \begin{aligned} & \int x^{2}(\ln x)^{2} d x=(\ln x)^{2} \int x^{2} d x-\int\left\{\frac{d}{d x}(\ln x)^{2} \int x^{2} \cdot d x\right\} d x \\ = & \frac{x^{3}(\ln x)^{2}}{3}-\frac{2}{3} \int \ln x^{2} d x \end{aligned} =∫x2(lnx)2dx=(lnx)2∫x2dx−∫{dxd(lnx)2∫x2⋅dx}dx3x3(lnx)2−32∫lnx2dx
=x3(lnx)23−23∫lnx⋅x2dx=x3(lnx)23−23[lnx∫x2dx−∫{ddx(lnx)∫x2dx]dx]=x3(lnx)23−2/3[x3lnx3−13∫x2dx]=x3(lnx)23−29x3lnx−29⋅13x3 \begin{array}{l} =\frac{x^{3}(\ln x)^{2}}{3}-\frac{2}{3} \int \ln x \cdot x^{2} d x \\ =\frac{x^{3}(\ln x)^{2}}{3}-\frac{2}{3}\left[\ln x \int x^{2} d x-\int\left\{\frac{d}{d x}(\ln x) \int x^{2} d x\right] d x\right] \\ =\frac{x^{3}(\ln x)^{2}}{3}-2 / 3\left[\frac{x^{3} \ln x}{3}-\frac{1}{3} \int x^{2} d x\right] \\ =\frac{x^{3}(\ln x)^{2}}{3}-\frac{2}{9} x^{3} \ln x-\frac{2}{9} \cdot \frac{1}{3} x^{3} \end{array} =3x3(lnx)2−32∫lnx⋅x2dx=3x3(lnx)2−32[lnx∫x2dx−∫{dxd(lnx)∫x2dx]dx]=3x3(lnx)2−2/3[3x3lnx−31∫x2dx]=3x3(lnx)2−92x3lnx−92⋅31x3
=x3{(lnx)23−29lnx−227}+c =x^{3}\left\{\frac{(\ln x)^{2}}{3}-\frac{2}{9} \ln x-\frac{2}{27}\right\}+c =x3{3(lnx)2−92lnx−272}+c
∫f(x)dx=x+e2x \int f{\left ( x \right )} dx = x + e^{2 x} ∫f(x)dx=x+e2x হলে f(x)=?
f(x)=x2 \mathrm{f(x)=x^{2}} f(x)=x2
∫xx(1+logx)dx=? \int x^{x} \left ( 1 + \log{x} \right ) dx = ? ∫xx(1+logx)dx=?
দৃশ্যকল্প-১: y=excos3x y=e^{x} \cos 3 x y=excos3x
দৃশ্যকল্প-২: চিত্রটি লক্ষ্য কর
