সৃজনশীলঃ ‘f(x)=ln⁡xx2+1……(i),g(x)=x^2+1……. (ii)$…’ রাজশাহী বোর্ড ২০১৭ – যোগজীকরণ, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

ধরি, $y=f\left(x\right)=\frac{\ln x}{x^2+1}$

$f^{\prime }\left(x\right)=\frac{(x^2+1)\frac{d}{dx}(\ln x)-\ln x\frac{d}{dx}(x^2+1)}{{(x^2+1)}^2}$

$=\frac{(x^2+1)\frac{1}{x}-\ln x\cdot 2x}{{(x^2+1)}^2}$

$=\frac{x^2+1-2x^2\ln x}{x}$

$=\frac{x^2+1-2x^2\ln x}{x{(x^2+1)}^2}$

$x=2$ বিन्দুতে $f^{\prime }\left(x\right)=\frac{2^2+1-2\times 2^2\times \ln 2}{2{(2^2+1)}^2}=\frac{5-8\ln 2}{50}$

$x=2$ হলে $y=\frac{\ln 2}{2^2+1}=\frac{\ln 2}{5}$

$(2,\frac{\ln 2}{5})$ বিন্দুতে স্পর্শকের সমীকরণ, 

$y-\frac{\ln 2}{5}=\frac{5-8\ln 2}{50}(x-2)$

বা, $50y-10\ln 2=(5-8\ln 2)x-10+16\ln 2$

$\therefore (5-8\ln 2)x-50y-10+26\ln 2=0\text{ (Ans.) }$

$\begin{array}{cc}& {\int }_0^{1}f\left(x\right)\cdot g\left(x\right)dx={\int }_0^1\frac{\ln x}{x^2+1}(x^2+1)dx\\ & ={\int }_0^1\ln xdx\\ & \therefore \int \ln xdx=\ln x\int dx-\int \left[\frac{d}{dx}\right(\ln x)\int dx]dx\\ & =x\ln x-\int \frac{1}{x}\cdot xdx\\ & =x\ln x-x+c\\ & \therefore {\int }_0^{1}f\left(x\right)g\left(x\right)dx=[x\ln x-x{]}_0^1\\ & =(0-1)-0=-1\text{ (Ans.) }\\ & \end{array}$