Question

СРОЧНО!!Вычислите интеграл, пожалуйста(Те примеры, которые обведены)

Answer 1

$1. \int_{0}^{2}\frac{dx}{3x+1}=\begin{vmatrix}t=3x+1\\ dx=\frac{1}{3}\end{vmatrix}=\frac{1}{3}\int_{1}^{7}\frac{dt}{t}=\left.\begin{matrix}\frac{1}{3}lnt\end{matrix}\right|_1^7=\frac{1}{3}ln7\\2. \int_{0}^{1}\left ( 1+\sqrt[3]{x} \right )dx=\left.\begin{matrix}\frac{3}{4}x^{\frac{4}{3}}\end{matrix}\right|^1_0=\frac{7}{4}\\3. \int_{-\pi}^{\pi}cos2xdx=\left.\begin{matrix}\frac{1}{2}sin2x\end{matrix}\right|_{-\pi}^{\pi=0}$

$4. \int_{1}^{2}\frac{x^3+3x^2+1}{x}dx=\int_{1}^{2}x^2dx+\int_{1}^{2}3xdx+\int_{1}^{2}\frac{dx}{x}=\left.\begin{matrix}\left ( \frac{x^3}{3}+\frac{3x^2}{2}+ln(|x|) \right )\end{matrix}\right|_1^2=\frac{41}{6}+ln(2)\\5. \int_{0}^{\pi}\frac{1-sin^2x}{cos^2x}dx=\int_{0}^{\pi}1dx=\left.\begin{matrix}x\end{matrix}\right|_0^{\pi}=\pi$