Question

Вычислить интегралы

Answer 1

$a)\int\limits^3_0 {\frac{1}{x+7} }\, dx=\ln(|x+7|)\bigg|_0^3=\ln(10)-\ln(7)=\ln(\frac{10}{7})\\b)\int\limits^6_3 {\frac{1}{x\ln(x)}} \, dx =\begin{vmatrix}\ t=\ln(x)\\\ dt=\frac{1}{x}dx\end{vmatrix}=\int\limits^6_3{\frac{dt}{t}}={\ln(|t|)}\bigg|_3^6=\ln(|\ln(x)|)\bigg|_3^6=\ln(\ln(6))-\ln(\ln(3))=\ln(\frac{\ln(6)}{\ln(3)})=\ln(\log_3(6))\\c)\int\limits^2_0 {(x-2)e^x} \, dx =\int\limits^2_0 {(xe^x-2e^x)} \, dx=xe^x-e^x-2e^x\bigg|_0^2=xe^x-3e^x\bigg|_0^2=\\=(2e^2-3e^2)-(0e^0-3e^0)=-e^2+3$