intigration of Rational Algebraic Fractions (মূলদ ভগ্নাংশ)
∫dxx+1+x−1=? \int \frac{d x}{\sqrt{x+1}+\sqrt{x-1}}= ? ∫x+1+x−1dx=?
13[(x−1)32−(x−1)32]+c\frac{1}{3}\left[(x-1)^{\frac{3}{2}}-(x-1)^{\frac{3}{2}}\right]+c 31[(x−1)23−(x−1)23]+c
13[(x+1)32+(x−1)32]+c\frac{1}{3}\left[(x+1)^{\frac{3}{2}}+(x-1)^{\frac{3}{2}}\right]+c 31[(x+1)23+(x−1)23]+c
13[(x+1)23−(x−1)23]+c\frac{1}{3}\left[(x+1)^{\frac{2}{3}}-(x-1)^{\frac{2}{3}}\right]+c 31[(x+1)32−(x−1)32]+c
13[(x+1)32−(x−1)32]+c\frac{1}{3}\left[(x+1)^{\frac{3}{2}}-(x-1)^{\frac{3}{2}}\right]+c 31[(x+1)23−(x−1)23]+c
Solve:
=∫x+1−x−1(x+1+x−1)(x+1−x−1)dx=∫x+1−x−1(x+1)−(x−1)dx=12∫[(x+1)1/2−(x+1)1/2]dx=12[(x+1)12+112+1−(x−1)12+112+1]+c=12[(x+1)3232−(x−1)3232]+c=13[(x+1)32−(x−1)32]+c (Ans:) \begin{array}{l} =\int \frac{\sqrt{x+1}-\sqrt{x-1}}{(\sqrt{x+1}+\sqrt{x-1})(\sqrt{x+1}-\sqrt{x-1})} d x \\ =\int \frac{\sqrt{x+1}-\sqrt{x-1}}{(x+1)-(x-1)} d x \\ =\frac{1}{2} \int\left[(x+1)^{1 / 2}-(x+1)^{1 / 2}\right] d x \\ =\frac{1}{2}\left[\frac{(x+1)^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{(x-1)^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]+c \\ =\frac{1}{2}\left[\frac{(x+1)^{\frac{3}{2}}}{\frac{3}{2}}-\frac{(x-1)^{\frac{3}{2}}}{\frac{3}{2}}\right]+c \\ =\frac{1}{3}\left[(x+1)^{\frac{3}{2}}-(x-1)^{\frac{3}{2}}\right]+c \text { (Ans:) } \end{array} =∫(x+1+x−1)(x+1−x−1)x+1−x−1dx=∫(x+1)−(x−1)x+1−x−1dx=21∫[(x+1)1/2−(x+1)1/2]dx=21[21+1(x+1)21+1−21+1(x−1)21+1]+c=21[23(x+1)23−23(x−1)23]+c=31[(x+1)23−(x−1)23]+c (Ans:)
∫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)
