intigration of Rational Algebraic Fractions (মূলদ ভগ্নাংশ)
∫dxx2−x+1 ∫ \frac{dx}{x^{2} - x + 1} ∫x2−x+1dx এর মান নির্ণয় কর-
13tan−1(2x−13)+C \frac{1}{\sqrt{3}} \tan^{- 1}{\left ( \frac{2 x - 1}{\sqrt{3}} \right )} + C 31tan−1(32x−1)+C
13tan−1(2x−12)+C \frac{1}{\sqrt{3}} \tan^{- 1}{\left ( \frac{2 x - 1}{\sqrt{2}} \right )} + C 31tan−1(22x−1)+C
∫dxx2−x+1=∫dxx2−x+14−14+1=∫dx(x−12)2+(32)2=132tan−1x−1232+c=23tan−1(2x−13)+c \int \frac{\mathrm{dx}}{\mathrm{x}^{2}-\mathrm{x}+1}=\int \frac{\mathrm{dx}}{\mathrm{x}^{2}-\mathrm{x}+\frac{1}{4}-\frac{1}{4}+1}=\int \frac{\mathrm{dx}}{\left(\mathrm{x}-\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}=\frac{1}{\frac{\sqrt{3}}{2}} \tan ^{-1} \frac{\mathrm{x}-\frac{1}{2}}{\frac{\sqrt{3}}{2}}+\mathrm{c}=\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \mathrm{x}-1}{\sqrt{3}}\right)+\mathrm{c} ∫x2−x+1dx=∫x2−x+41−41+1dx=∫(x−21)2+(23)2dx=231tan−123x−21+c=32tan−1(32x−1)+c
P=(x−4)2(x−3) P=(x-4)^{2}(x-3) P=(x−4)2(x−3) এবং g(x,y)=x2+y2 g(x, y)=x^{2}+y^{2} g(x,y)=x2+y2
দৃশ্যকল্প: g(x)=cot−1x,f(x)=x g(x)=\cot ^{-1}x, f(x)=x g(x)=cot−1x,f(x)=x
P=(x−4)2(x−3) P=(x-4)^{2}(x-3) P=(x−4)2(x−3)
f(x)=x2 f(x)=x^{2} f(x)=x2 এবং x2a2+y2b2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 a2x2+b2y2=1
দৃশ্যকল্প-১: f(x)=x+3(x−1)(x2+5)f(x)=\frac{x+3}{(x-1)\left(x^{2}+5\right)}f(x)=(x−1)(x2+5)x+3.
দৃশ্যকল্প-২: y2=8x,x−y=0\mathrm{y}^{2}=8 \mathrm{x}, \mathrm{x}-\mathrm{y}=0y2=8x,x−y=0.
