第8章 8.4

(1)

$\int \log(2x)dx = x \cdot \log(2x) - \int x \cdot \frac{2}{2x} dx = x \cdot \log(2x) - \int dx = x \cdot \log(2x) - x$

(2)

$\int xe^{2x}dx = x \cdot \frac{1}{2}e^{2x} - \frac{1}{2}\int e^{2x}dx = \frac{x}{2}e^{2x} - \frac{1}{2} \cdot \frac{1}{2}e^{2x} = \frac{x}{2}e^{2x} - \frac{1}{4}e^{2x}$

(3)

$\int \sqrt{x} \log x dx = \frac{2}{3}\sqrt{x^3} \log x - \frac{2}{3}\int \frac{\sqrt{x^3}}{x} dx = \frac{2}{3}\sqrt{x^3} \log x - \frac{2}{3}\int \sqrt{x} dx = \frac{2}{3}\sqrt{x^3} \log x - \frac{4}{9}\sqrt{x^3}$

(4)

$\int x^2 e^x dx = x^2 e^x - \int 2xe^x dx = x^2 e^x - 2xe^x + 2\int e^x dx = x^2 e^x - 2xe^x + 2e^x = e^x(x^2 - 2x + 2)$

(5)

$\int x\cos 2x \, dx = x \cdot \frac{1}{2}\sin 2x - \frac{1}{2}\int \sin 2x \, dx = \frac{x}{2}\sin 2x + \frac{1}{4}\cos 2x$

(6)

$\int x\sin 3x \, dx = x \cdot \left(-\frac{1}{3}\right)\cos 3x + \frac{1}{3}\int \cos 3x \, dx = -\frac{x}{3}\cos 3x + \frac{1}{9}\sin 3x$

(7)

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

(8)

$\int e^{-x}\sin x \, dx = -e^{-x}\sin x + \int e^{-x}\cos x \, dx = -e^{-x}\sin x - e^{-x}\cos x - \int e^{-x}\sin x \, dx$

$2\int e^{-x} \sin x dx = -e^{-x} \sin x - e^{-x} \cos x = -e^{-x} (\sin x + \cos x)$

$\int e^{-x} \sin x dx = -\frac{1}{2} e^{-x} (\sin x + \cos x)$