4.4 三类常见可积函数积分 | 花野猫的数字花园

4.4 三类常见可积函数积分 | 花野猫的数字花园

1

+

u

)

+

C

= \frac{1}{8} \ln \frac{1 - \cos x}{1 + \cos x} + \frac{1}{4 (1 + \cos x)} + C .

\begin{aligned} \text{原式} &=\int\frac{\mathrm{d}x}{2\sin x(\cos x+1)}=\int\frac{\sin x\mathrm{d}x}{2(1-\cos^2x)(1+\cos x)}\\ &\xlongequal{\cos x=u}-\frac12\int\frac{\mathrm{d}u}{(1-u^2)(1+u)} =-\frac14\int\left[\frac{1}{1-u^2}+\frac{1}{(1+u)^2}\right]\mathrm{d}u\\ &=\frac18\left(\ln\frac{1-u}{1+u}+\frac{2}{1+u}\right)+C\\ &=\frac18\ln\frac{1-\cos x}{1+\cos x}+\frac{1}{4(1+\cos x)}+C. \end{aligned}

+

\frac{1}{2}

\int

\frac{1}{1 - u^{2}}

d

u

=

-

\frac{1}{2}

\frac{1}{u + 1}

-

\frac{1}{4}

\ln

∣ \frac{u - 1}{u + 1} ∣

+

C

= - \frac{1}{2} \frac{1}{\sin x + 1} - \frac{1}{4} \ln ∣ \frac{\sin x - 1}{\sin x + 1} ∣ + C .

\begin{aligned} \text{原式} &=\int\frac{\cos x\mathrm{d}x}{(1-\sin^2x)(1+\sin x)} =\int\frac{\mathrm{d}(\sin x)}{(1-\sin^2x)(1+\sin x)}\\ &\xlongequal{\sin x=u}\int\frac{\mathrm{d}u}{(1-u^2)(1+u)} =\frac12\int\frac{1-u+1+u}{(1-u^2)(1+u)}\mathrm{d}u &=\frac12\int \frac{1}{(1+u)^2}\mathrm{d}u+\frac12\int \frac{1}{1-u^2}\mathrm{d}u =-\frac12\frac{1}{u+1}-\frac14\ln\left|\frac{u-1}{u+1}\right|+C\\ &=-\frac12\frac{1}{\sin x+1}-\frac14\ln\left|\frac{\sin x-1}{\sin x+1}\right|+C. \end{aligned}

x

=

\frac{1}{t^{2} - 1}

,

d

x

=

-

\frac{2 t}{(t^{2} - 1)^{2}}

d

t

\displaystyle x=\frac{1}{t^2-1},\mathrm{d}x=-\frac{2t}{(t^2-1)^2}\mathrm{d}t

\ln

∣ x + \frac{1}{2} + \sqrt{x^{2} + x} ∣

-

2

\sqrt{1 + \frac{1}{x}}

+

C

.

\begin{aligned} \int \frac1x\sqrt{\frac{x+1}{x}}\mathrm{d}x &=\int \frac{x+1}{x\sqrt{x^2+x}}\mathrm{d}x\\ &=\int \frac{\mathrm{d}\left(x+\frac12\right)}{\sqrt{\left(x+\frac12\right)^2-\frac14}}+\int \frac{\mathrm{d}x}{x^2\sqrt{1+\frac1x}}\\ &=\ln\left|x+\frac12+\sqrt{\left(x+\frac12\right)^2-\frac14}\right|-\int \frac{\mathrm{d}\left(1+\frac1x\right)}{\sqrt{1+\frac1x}}\\ &=\ln\left|x+\frac12+\sqrt{x^2+x}\right|-2\sqrt{1+\frac1x}+C. \end{aligned}