Question

Нужно решить два неопределенных и два определенных интеграла.

Answer 1

$LARGE \ intbegin{pmatrix} 3x^2-2cos{x} end{pmatrix}mathrm{dx}=3int x^2mathrm{dx}-2intcos{x}mathrm{dx}=x^3-2sin{x}+C\ int{2xmathrm{dx}over1+x^2}={2over2}int{mathrm{d(1+x^2)}over1+x^2}=ln{|1+x^2|}+C\ int_{2}^{8}{2+xover x^2}mathrm{dx}=2int_{2}^{8}{mathrm{dx}over x^2}+int_{2}^{8}{xmathrm{dx}over x^2}=-{2over x}|_{2}^{8}+ln{(|x|)}|_{2}^{8}=-2({1over8}-{1over2})+ln{8}-ln{2}={3over4}+ln{4}\ int_{piover6}^{piover2}{cos{x}over1+sin{x}}mathrm{dx}=int{mathrm{d(1+sin{x})}over1+sin{x}}=ln{(|1+sin{x}|)}|_{piover6}^{piover2}=ln{2}-ln{1,5}=ln{4over3}$