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1.18 Simple examples

To find (1) $\int{{x^{2}+1}\over x}\,dx$ and (2) $\int{{x+2}\over{x+1}}\,dx$ .

Solution.

For (1), we have $\int{{x^{2}+1}\over x}\,dx={\int(x+{1\over x})\,dx={{x^{2}}\over 2}+\log|x|+c}$ for some constant $c$ .

For (2), we express ${{x+2}\over{x+1}}$ as ${{(x+1)+1}\over{x+1}}=1+{{1}\over{x+1}}$ .

Hence $\int{{x+2}\over{x+1}}dx=$ $\int\left(1+\frac{1}{x+1}\right)\,dx=x+\log|x+1|+c$ for some constant $c$ .