Question

Punctul b.
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Answer 1

$\displaystyle \mathtt{2.~~~f:\mathbb{R}\rightarrow\mathbb{R},~f(x)=x^3+x^2+x+1}\\\\ \mathtt{b)~\int\limits_{-1}^1 \frac{f(x)}{x^2+1}dx=2~~~~~~~~~~~~~~~~~~~\int\limits_{-1}^1\frac{x^3+x^2+x+1}{x^2+1}dx=2}\\\\\mathtt{x^3+x^2+x+1=x^2(x+1)+(x+1)=(x+1)(x^2+1)}\\\\ \mathtt{\int\limits_{-1}^1 \frac{(x+1)(x^2+1)}{x^2+1}dx=\int\limits_{-1}^1(x+1)dx=\int\limits_{-1}^1xdx+\int\limits_{-1}^11dx=}\\ \\\mathtt{=\frac{x^2}{2}\Bigg|_{-1}^1+x\Bigg|_{-1}^1=\frac{1^2}{2}-\frac{(-1)^2}{2}+1-(-1)=\frac{1}{2}-\frac{1}{2}+1+1=2}$