intigration of Rational Algebraic Fractions (মূলদ ভগ্নাংশ)
∫1x+x+1dx\int\dfrac{1}{\sqrt{x}+\sqrt{x+1}}dx∫x+x+11dx is equal to
23[(x+1)3/2−x3/2]+C\dfrac{2}{3}[(x+1)^{3/2}-x^{3/2}]+C32[(x+1)3/2−x3/2]+C
2x3[x+1−x]+C\dfrac{2x}{3}[\sqrt{x+1}-\sqrt{x}]+C32x[x+1−x]+C
ln[x−x+1]+C\ln[\sqrt{x}-\sqrt{x+1}]+Cln[x−x+1]+C
ln[x+1−x]+C\ln[\sqrt{x+1}-\sqrt{x}]+Cln[x+1−x]+C
I=∫1x+x+1dxI=∫1x+x+1×x−x+1x−x+1dxI=∫x−x+1x−x−1dxI=−1[∫xdxI=−∫x+1dx]I=−[x3/232−(x+1)3/232]+cI=23[(x+1)3/2−x3/2]+c\begin{array}{l} I=\int { \dfrac { 1 }{ { \sqrt { x } +\sqrt { x+1 } } } dx } \\I= \int { \dfrac { 1 }{ { \sqrt { x } +\sqrt { x+1 } } } \times \dfrac { { \sqrt { x } -\sqrt { x+1 } } }{ { \sqrt { x } -\sqrt { x+1 } } } dx } \\I= \int { \dfrac { { \sqrt { x } -\sqrt { x+1 } } }{ { x-x-1 } } dx } \\ I=-1\left[ { \int { \sqrt { x } dx } I=-\int { \sqrt { x+1 } dx } } \right] \\ I=-\left[ { \dfrac { { { x^{ 3/2 } } } }{ { \dfrac { 3 }{ 2 } } } -\dfrac { { { { \left( { x+1 } \right) }^{ 3/2 } } } }{ { \dfrac { 3 }{ 2 } } } } \right] +c \\ I=\dfrac { 2 }{ 3 } \left[ { { { \left( { x+1 } \right) }^{ 3/2 } }-{ x^{ 3/2 } } } \right] +c \end{array}I=∫x+x+11dxI=∫x+x+11×x−x+1x−x+1dxI=∫x−x−1x−x+1dxI=−1[∫xdxI=−∫x+1dx]I=−23x3/2−23(x+1)3/2+cI=32[(x+1)3/2−x3/2]+c
∫x2+x+1x2−x+1dx \int \frac{x^{2}+x+1}{x^{2}-x+1} d x ∫x2−x+1x2+x+1dx
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
