第8章 8.3

(1)

$\int (5-6x)^3\,dx = \frac{1}{4}\cdot\left(-\frac{1}{6}\right)(5-6x)^4 = -\frac{1}{24}(5-6x)^4$

(2)

$\int \sqrt{4x-2}\,dx = \frac{2}{3}\cdot\frac{1}{4}\sqrt{(4x-2)^3} = \frac{1}{6}\sqrt{(4x-2)^3}$

(3)

$\int \sin\frac{\pi}{3}x\,dx = -\frac{3}{\pi}\cos\frac{\pi}{3}x$

(4)

$\int 2x(x^2-3)^3\,dx \qquad x^2 = t \text{ とおくと} \qquad 2x\,dx = dt \qquad \text{これより}$

$\int (t-3)^3\,dxt = \frac{1}{4}(t-3)^4 \qquad \text{元に戻して} \qquad \int 2x(x^2-3)^3\,dx = \frac{1}{4}(x^2-3)^4$

(5)

$\int \frac{x^2}{\sqrt{2x^3-1}}\,dx \qquad x^3 = t \text{ とおくと} \qquad 3x^2\,dx = dt \qquad \text{これから}$

$\int \frac{1}{\sqrt{2t-1}}\cdot\frac{1}{3}\,dt = \frac{1}{3}\sqrt{2t-1} \qquad \text{元に戻して} \qquad \int \frac{x^2}{\sqrt{2x^3-1}}\,dx = \frac{1}{3}\sqrt{2x^3-1}$

(6)

$\int \frac{1}{x}\log x\,dx \qquad \log x = t \text{ とおくと} \qquad \frac{1}{x}dx = dt \qquad \text{これから}$

$\int t\,dt = \frac{1}{2}t^2 \qquad \text{元に戻して} \qquad \int \frac{1}{x}\log x\,dx = \frac{1}{2}(\log x)^2$

(7)

$\int \frac{1}{\tan x}\,dx = \int \frac{\cos x}{\sin x}\,dx$

$\sin x = t$ とおくと $\cos x\,dx = dt$ これから

$\int \frac{1}{t}\,dt = \log|t|$

元に戻して

$\int \frac{1}{\tan x}\,dx = \log|\sin x|$

(8)

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