Maths

$\begin{array}{l}Let,\; I\; ={\displaystyle \int \frac{x-3}{{(x-1)}^3}}{e}^{x}dx.\\ ={\displaystyle \int \frac{(x-1)-2}{{(x-1)}^3}}{e}^{x}dx.\\ ={\displaystyle \int e^x}\left[\frac{x-1}{{(x-1)}^3}-\frac{2}{{(x-1)}^3}\right]dx\\ ={\displaystyle \int e^x}\left[\frac{1}{{(x-1)}^2}+\frac{(-2)}{{(x-1)}^3}\right]dx,whichisinfore\\ {\displaystyle \int e^x}\left[f(x)+f^{\prime }(x)\right]dx,Where\\ f(x)=\frac{1}{{(x-1)}^2}\\ f^{\prime }(x)=\frac{d}{dx}{(x-1)}^{-2}=\frac{-2}{{(x-1)}^3}.\\ \therefore I=e^{x}f(x)+c=e^x\frac{1}{{(x-1)}^2}+c=\frac{e^x}{{(x-1)}^2}+\; C\end{array}$

$\begin{array}{l}Let,\; I={\displaystyle \int e^x}\left(\frac{1}{x}-\frac{1}{x^2}\right)dx{\displaystyle \int e^x}[f(x)+f(x)]dx\\ Where,f(x)=\frac{1}{x}\\ f(x)=\frac{dx}{dx}=-1x^{-2}=\frac{-1}{x^2}\end{array}$

So, I = e^xf (x) + C

$\begin{array}{l}=e^x\frac{1}{x}+\; c\\ =\frac{e^x}{x}+\; c\end{array}$

$Let,={\displaystyle \int e^x}\left(\frac{1+sinx}{1+cosx}\right)dx.,$

$={\displaystyle \int e^x}\frac{1+2sin\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.cos\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{2co{s}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}dx.$ {∴ sin 2x = 2sin x cos x. cos 2x = cos^{- 2}x- 1 1 + cos 2x = 2cos²x.

$\begin{array}{l}={\displaystyle \int e^x}\left[\frac{1}{2co{s}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\frac{2sin\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.cos\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{2co{s}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]dx]\\ ={\displaystyle \int e^x}\left[\frac{1}{2}se{c}^2\frac{x}{2}+\frac{cosx\cdot sin\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{cos\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]dx\end{array}$

$={\displaystyle \int e^x}\left[tan\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\frac{1}{2}si{n}^2\frac{x}{2}\right]dx$ is in the form

  e^x [f(x) + f(x)].dx where

$\begin{array}{l}f(x)=tan\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ f^{\prime }(x)=\frac{d}{dr}tan\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=se{c}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\frac{d}{dx}\left(\frac{x}{2}\right)\\ =\frac{1}{2}se{c}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ =e^{x}f(x)+c=e^{x}tan\frac{x}{2}+\; C\end{array}$

$Let,I={\displaystyle \int \frac{x{e}^x}{{(x+1)}^2}}dx={\displaystyle \int \frac{x+1-1}{{(x+1)}^2}}{e}^{x}dx$

$={\displaystyle \int e^x}\left[\frac{(x+1)}{{(x+1)}^2}+\frac{(-1)}{{(x+1)}^2}\right]dx.$ is of the form

e^x [f(x) + f(x)] dx

$\begin{array}{l}Where,f(x)=\frac{x+1}{{(x+1)}^2}=\frac{1}{x+1}\\ So,f^{\prime }(x)=\frac{d{(x+1)}^{-1}}{dx}=\frac{(-1)}{{(x+1)}^2}.\\ \therefore {\displaystyle \int \frac{x^{\cdot }\cdot e^x}{\left(1+x^2\right)}}dx=e^x\left[\frac{1}{x+1}\right]+\; C\end{array}$

∴f (x) = sin x

f (x) = cos x.

e^x [f (x) + f (x)] dx = e^xf (x) + C

${\displaystyle \int e^x(sinx+cosx)dx=e^x}sinx+c$

$\begin{array}{l}{\displaystyle \int \left(x^2+1\right)}logxdx=logx{\displaystyle \int \left(x^2+1\right)dx-{\displaystyle \int \left(x^2+1\right)dx-{\displaystyle \int \frac{d}{dx}logx{\displaystyle \int \left(x^2+1\right)dxdx}}}}\\ =logx\cdot \left[\frac{x^3}{3}+x\right]-{\displaystyle \int \frac{1}{x}*\left[\frac{x^3}{3}+x\right]dx}\\ =\left[\frac{x^3}{3}+x\right]logx-{\displaystyle \int \left[\frac{x^2}{3}+1\right]dx}\\ =\left[\frac{x^3}{3}+x\right]logx-\frac{x^3}{3*3}-x+\; C\\ =\left[\frac{x^3}{3}+x\right]logx-\frac{x^3}{9}-x+\; C\end{array}$

$\begin{array}{l}{\displaystyle \int x{\left(logx\right)}^2=\cdot {\left(logx\right)}^2{\displaystyle \int xdx}-{\displaystyle \int \frac{d}{dx}{\left(logx\right)}^2\cdot {\displaystyle \int xdxdx}}}\\ ={\left(logx\right)}^2*\frac{x^2}{2}-{\displaystyle \int 2logx*\frac{1}{2}*\frac{x^2}{2}dx}\\ =\frac{x^2}{2}{\left(logx\right)}^2-{\displaystyle \int logx\cdot xdx}\end{array}$

$\begin{array}{l}=\frac{x^2}{2}{\left(logx\right)}^2-\left[logx{\displaystyle \int xdx-{\displaystyle \int \frac{d}{dx}logx{\displaystyle \int xdxdx}}}\right]\\ =\frac{x^2}{2}{\left(logx\right)}^2-\frac{x^2}{2}logx+{\displaystyle \int \frac{x}{2}dx}\\ =\frac{x^2}{2}{\left(logx\right)}^2-\frac{x^2}{2}logx+\frac{x^2}{4}+\; C\end{array}$

$\begin{array}{l}{\displaystyle \int ta{n}^{-1}xdx={\displaystyle \int \left(ta{n}^{-1}x\right)1\cdot dx}}.\\ =ta{n}^{-1}x{\displaystyle \int dx-{\displaystyle \int \frac{d}{dx}ta{n}^{-1}x{\displaystyle \int dxdx}}}\\ =xta{n}^{-1}x-{\displaystyle \int \frac{1}{1+x^2}\cdot xdx}.\\ =xta{n}^{-1}x-\frac{1}{2}{\displaystyle \int \frac{2x}{1+x^2}dx}\\ =xta{n}^{-1}x-\frac{1}{2}\cdot log\left|1+x^2\right|+\; C\end{array}$

$\begin{array}{l}{\displaystyle \int xse{c}^{2}xdx=x{\displaystyle \int se{c}^{2}xdx-{\displaystyle \int \frac{dx}{dx}{\displaystyle \int se{c}^{2}xdx}dx.}}}\\ =xtanx-{\displaystyle \int tanxdx}\\ =xtanx-\left(-log\left|cosx\right|\right)+\; C\\ =xtanx+log\left|cosx\right|+\; C\end{array}$

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