intigration of Rational Algebraic Fractions (মূলদ ভগ্নাংশ)
∫1(1−4x)3dx=\int \frac{1}{\sqrt[3]{(1-4 x)}} d x = ∫3(1−4x)1dx=
−38(1−4x)32+c-\frac{3}{8}(1-4 x)^{\frac{3}{2}}+c−83(1−4x)23+c
−38(1+4x)+c-\frac{3}{8}(1+4 x)+c−83(1+4x)+c
−38(1−4x)23+c-\frac{3}{8}(1-4 x)^{\frac{2}{3}}+c−83(1−4x)32+c
−38(1+4x)23+c-\frac{3}{8}(1+4 x)^{\frac{2}{3}}+c−83(1+4x)32+c
Solve:
∫1(1−4x)3dx=∫1(1−4x)1/3dx=∫(1−4x)−13dx=(1−4x)13+1(−13+1)(−4)+c=(1−4x)2323(−4)+c=−38(1−4x)23+c \begin{aligned} & \int \frac{1}{\sqrt[3]{(1-4 x)}} d x=\int \frac{1}{(1-4 x)^{1 / 3}} d x \\ = & \int(1-4 x)^{-\frac{1}{3}} d x=\frac{(1-4 x)^{\frac{1}{3}+1}}{\left(-\frac{1}{3}+1\right)(-4)}+c \\ = & \frac{(1-4 x)^{\frac{2}{3}}}{\frac{2}{3}(-4)}+c=-\frac{3}{8}(1-4 x)^{\frac{2}{3}}+c \end{aligned} ==∫3(1−4x)1dx=∫(1−4x)1/31dx∫(1−4x)−31dx=(−31+1)(−4)(1−4x)31+1+c32(−4)(1−4x)32+c=−83(1−4x)32+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
∫1ex+1dx=?\int \frac{1}{e^{x}+1} d x = ?∫ex+11dx=?
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)
