Evaluate the following integrals:

Question:

Evaluate $\int \frac{1}{1+x+x^{2}+x^{3}} d x$

Solution:

$=\int \frac{d x}{1+x+x^{2}+x^{3}}=\int \frac{d x}{(1+x)\left(1+x^{2}\right)}$

We can write the integral as follows,

$=\int\left[\frac{d x}{2(x+1)}\right]-\int\left[\frac{x-1}{2\left(x^{2}+1\right)} d x\right]=\frac{1}{2} \log (x+1)-\frac{1}{2}\left[\int \frac{x d x}{x^{2}+1}-\int \frac{d x}{x^{2}+1}\right]$

$=\frac{1}{2} \log (x+1)-\frac{1}{2}\left[\log \frac{\left(x^{2}+1\right)}{2}-\tan ^{-1} x\right]+c$

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