Question

Решить с 1 по 6. Легкие баллы

Answer 1

Ответ:

$1)\ \ \int \Big (\dfrac{x^2}{3}+\dfrac{x^{15}}{4}+\dfrac{x}{12}\Big)\, dx=\dfrac{x^3}{9}+\dfrac{x^{16}}{64}+\dfrac{x^2}{24}+C\\\\\\2)\ \ \int (x+1)(x-\sqrt{x}+1)\, dx=\int (x^2+2x+1-x^{3/2}-x^{1/2})\, dx=\\\\=\dfrac{x^3}{3}+x^2+x-\dfrac{2x^{5/2}}{5}-\dfrac{2x^{3/2}}{3}+C\\\\\\3)\ \ \int \sqrt[8]{(8x)^{-7}^}\, dx=\int (8x)^{-7/8}\, dx=\dfrac{1}{8^{7/8}}\cdot \dfrac{x^{1/8}}{1/8}+C=\sqrt[8]8\cdot \sqrt[8]{x}}+C=\sqrt[8]{8x}+C$

$4)\ \ \int \dfrac{dx}{x^2+7}=\dfrac{1}{\sqrt7}\cdot arctg\dfrac{x}{\sqrt7}+C\\\\\\5)\ \ \int \dfrac{3x^3-2x^2}{8x}\, dx=\int \Big(\dfrac{3x^2}{8}-\dfrac{x}{4}\Big)\, dx=\dfrac{x^3}{8}-\dfrac{x^2}{8}+C\\\\\\6)\ \ \int \dfrac{dx}{x^2-10}=\dfrac{1}{2\sqrt{10}}\cdot ln\Big|\dfrac{x-\sqrt{10}}{x+\sqrt{10}}\Big|+C$