第8章 8.2

(1)

$\int \left(\sqrt[3]{x} + \frac{1}{\sqrt[3]{x^2}}\right) dx = \int \left(x^{\frac{1}{3}} + x^{-\frac{2}{3}}\right) dx = \frac{3}{4}x^{\frac{4}{3}} + 3x^{\frac{1}{3}} + C$

(2)

$\int \left(2e^{x} - \frac{1}{x}\right) dx = 2e^{x} - \log x + C$

(3)

$\int \left(\cos x + \frac{1}{\sin^2 x}\right) dx = \sin x - \frac{1}{\tan x} + C$

(4)

$\int \tan^2 x\, dx = \int \left(\frac{\sin^2 x}{\cos^2 x}\right) dx = \int \left(\frac{1 - \cos^2 x}{\cos^2 x}\right) dx = \int \left(\frac{1}{\cos^2 x} - 1\right) dx = \tan x - x + C$

(5)

$\int \left(\frac{(1+x)^2}{\sqrt[3]{x^2}}\right) dx = \int \left(\frac{1 + 2x + x^2}{x^{\frac{2}{3}}}\right) dx = \int \left(x^{-\frac{2}{3}} + 2x^{\frac{1}{3}} + x^{\frac{4}{3}}\right) dx = 3x^{\frac{1}{3}} + 2 \cdot \frac{3}{4}x^{\frac{4}{3}} + \frac{3}{7}x^{\frac{7}{3}} + C$ $= 3x^{\frac{1}{3}} + \frac{3}{2}x^{\frac{4}{3}} + \frac{3}{7}x^{\frac{7}{3}} + C$

(6)

$\int \left(\frac{\cos^2 x}{\sin^2 x}\right) dx = \int \left(\frac{1 - \sin^2 x}{\sin^2 x}\right) dx = \int \left(\frac{1}{\sin^2 x} - 1\right) dx = -\frac{1}{\tan x} - x + C$