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Evaluate:

Question:

Evaluate: $\int \frac{\mathrm{x}^{2}+3 \mathrm{x}-1}{(\mathrm{x}+1)^{2}} \mathrm{dx}$

Solution:

$\Rightarrow \int \frac{x^{2}+x+2 x-1}{(x+1)^{2}} d x$

$\Rightarrow \int \frac{x(x+1)+2 x-1}{(x+1)^{2}} d x$

$\Rightarrow \int \frac{x(x+1)}{(x+1)^{2}} d x+\int \frac{2 x-1}{(x+1)^{2}} d x$

$\Rightarrow \int \frac{x}{x+1} d x+\int \frac{2 x+2-2-1}{(x+1)^{2}} d x$

$\Rightarrow \int \frac{x+1-1}{x+1} d x+\int \frac{2(x+1)-3}{(x+1)^{2}} d x$

$\Rightarrow \int d x-\int \frac{1}{x+1} d x+\int \frac{2}{x+1} d x-\int \frac{3}{(x+1)^{2}} d x$

We know $\int \mathrm{x} \mathrm{dx}=\frac{\mathrm{x}^{\mathrm{n}}}{\mathrm{n}+1} ; \int \frac{1}{\mathrm{x}} \mathrm{dx}=\ln \mathrm{x}$

$\Rightarrow \mathrm{x}-\ln (\mathrm{x}+1)+2 \ln (\mathrm{x}+1)-\int 3(\mathrm{x}+1)^{-2} \mathrm{dx}$

$\Rightarrow \mathrm{x}-\ln (\mathrm{x}+1)+2 \ln (\mathrm{x}+1)+\frac{3}{\mathrm{x}+1}+c$

$\Rightarrow \mathrm{x}+\ln (\mathrm{x}+1)+\frac{3}{\mathrm{x}+1}+\mathrm{c} .$ (Where $\mathrm{c}$ is some arbitrary constant)

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