Question

ПОМОГИТЕ решить определенные интегралы!!

Answer 1

$\LARGE \\ \int_{1}^{6}{x\over\sqrt{x+3}}\mathrm{dx}=\begin{vmatrix} x+3=t^2, \; t_1=2\\ \mathrm{dx}=2t\mathrm{dt}, \; t_2=3 \end{vmatrix}=2\int_{2}^{3}{t(t^2-3)\mathrm{dt}\over t}=2\int_{2}^{3}t^2\mathrm{dt}-6\int_{2}^{3}\mathrm{dt}={2\over3}t^3|_{2}^{3}-6t|_{2}^{3}={2\over3}(27-8)-6(3-2)={38\over3}-{18\over3}=\boldsymbol{{20\over3}}\\\\ \int_{0}^{1}xe^x\mathrm{dx}=\begin{vmatrix} u=x, \mathrm{du}=\mathrm{dx}\\ \mathrm{dv}=e^x\mathrm{dx}, v=e^x \end{vmatrix}=xe^x|_{0}^{1}-\int_{0}^{1}e^x\mathrm{dx}=(1\cdot e^1-0\cdot e^0)-e^x|_{0}^{1}=e-(e^1-e^0)=\boldsymbol{1}$