UV আকারের (Integration by parts)
∫xnlnxdx=? \int x^{n} \ln x d x = ?∫xnlnxdx=?
−xn+1n+1lnx+xn+1(n+1)2+c-\frac{x^{n+1}}{n+1} \ln x+\frac{x^{n+1}}{(n+1)^{2}}+c−n+1xn+1lnx+(n+1)2xn+1+c
xn+1n+1lnx+xn+1(n+1)2+c\frac{x^{n+1}}{n+1} \ln x+\frac{x^{n+1}}{(n+1)^{2}}+cn+1xn+1lnx+(n+1)2xn+1+c
−xn+1n+1lnx−xn+1(n+1)2+c-\frac{x^{n+1}}{n+1} \ln x-\frac{x^{n+1}}{(n+1)^{2}}+c−n+1xn+1lnx−(n+1)2xn+1+c
xn+1n+1lnx−xn+1(n+1)2+c\frac{x^{n+1}}{n+1} \ln x-\frac{x^{n+1}}{(n+1)^{2}}+cn+1xn+1lnx−(n+1)2xn+1+c
Solve:
=lnx∫xndx−∫{ddx(lnx)∫xndx}dx=lnx⋅xn+1n+1−∫1x⋅xn+1n+1dx=xn+1n+1lnx−1n+1∫xndx=xn+1n+1lnx−1n+1⋅xn+1n+1+c=xn+1n+1lnx−xn+1(n+1)2+c \begin{array}{l} =\ln x \int x^{n} d x-\int\left\{\frac{d}{d x}(\ln x) \int x^{n} d x\right\} d x \\ =\ln x \cdot \frac{x^{n+1}}{n+1}-\int \frac{1}{x} \cdot \frac{x^{n+1}}{n+1} d x \\ =\frac{x^{n+1}}{n+1} \ln x-\frac{1}{n+1} \int x^{n} d x \\ =\frac{x^{n+1}}{n+1} \ln x-\frac{1}{n+1} \cdot \frac{x^{n+1}}{n+1}+c \\ =\frac{x^{n+1}}{n+1} \ln x-\frac{x^{n+1}}{(n+1)^{2}}+c \end{array} =lnx∫xndx−∫{dxd(lnx)∫xndx}dx=lnx⋅n+1xn+1−∫x1⋅n+1xn+1dx=n+1xn+1lnx−n+11∫xndx=n+1xn+1lnx−n+11⋅n+1xn+1+c=n+1xn+1lnx−(n+1)2xn+1+c
Ai এর মাধ্যমে
১০ লক্ষ+ প্রশ্ন ডাটাবেজ
প্র্যাকটিস এর মাধ্যমে নিজেকে তৈরি করে ফেলো
উত্তর দিবে তোমার বই থেকে ও তোমার মত করে।
সারা দেশের শিক্ষার্থীদের মধ্যে নিজের অবস্থান যাচাই
The sequence S0,S1,S2S_0,S_1,S_2S0,S1,S2.... forms a G.P with common ratio
If ∫f(x)dx=g(x),\displaystyle\int f(x)dx=g(x),∫f(x)dx=g(x), then ∫f−1(x)dx=\displaystyle\int f^{-1}(x)dx=∫f−1(x)dx= _____________+++c.
The integrating factor of the differential equation dydx(xloge x)+y=2loge x\dfrac{dy}{dx}\left(x\log _e\:x\right)+y=2\log _e\:xdxdy(xlogex)+y=2logex is given by
∫e10xsin6xdx= \int e^{10 x} \sin{6} x dx = ∫e10xsin6xdx= কত?
