DEFINITE INTEGRALS CONTAINING LOGARITHMIC FUNCTIONS

1.

$$\int_{0}^{1}x^m(\ln x)^n dx=\displaystyle \frac{(-1)^n n!}{(m+1)^{n+1}}\hspace{.2in}m>-1, n=0,1,2,...$$

2.

$$\int_{0}^{1}\displaystyle \frac{\ln x}{1+x}dx=-\displaystyle \frac{\pi^2}{12}$$

3.

$$\int_{0}^{1}\displaystyle \frac{\ln x}{1-x}dx=-\displaystyle \frac{\pi^2}{6}$$

4.

$$\int_{0}^{1}\displaystyle \frac{\ln(1+x)}{x}dx=\displaystyle \frac{\pi^2}{12}$$

5.

$$\int_{0}^{1}\displaystyle \frac{\ln(1-x)}{x}dx=-\displaystyle \frac{\pi^2}{6}$$

6.

$$\int_{0}^{1}\ln x \ln(1+x)dx=2-2\ln 2-\displaystyle \frac{\pi^2}{12}$$

7.

$$\int_{0}^{1}\ln x\ln(1-x)dx=2-\displaystyle \frac{\pi^2}{6}$$

8.

$$\int_{0}^{\infty}\displaystyle \frac{x^{p-1}\ln x}{1+x}dx=-\pi^2\csc p\pi \cot p\pi \hspace{.2in}0<p<1$$

9.

$$\int_{0}^{1}\displaystyle \frac{x^m-x^n}{\ln x}dx=\ln\displaystyle \frac{m+1}{n+1}$$

10.

$$\int_{0}^{\infty}e^{-x}\ln x dx=-\gamma$$

where the constant $\gamma$ is the Euler's constant.

11.

$$\int_{0}^{\infty}e^{-x^2}\ln x dx=-\displaystyle \frac{\displaystyle \sqrt{\pi}}{4}(\gamma+2\ln 2)$$

where the constant $\gamma$ is the Euler's constant.

12.

$$\int_{0}^{\infty}\ln\left(\displaystyle \frac{e^x+1}{e^x-1}\right)dx=\displaystyle \frac{\pi^2}{4}$$

13.

$$\int_{0}^{\pi/2}\ln\sin x dx=\int_{0}^{\pi/2}\ln\cos x dx=-\displaystyle \frac{\pi}{2}\ln2$$

14.

$$\int_{0}^{\pi/2}(\ln\sin x)^2 dx=\int_{0}^{\pi/2}(\ln\cos x)^2 dx=\displaystyle \frac{\pi}{2}(\ln 2)^2+\displaystyle \frac{\pi^3}{24}$$

15.

$$\int_{0}^{\pi}x\ln\sin x dx=-\displaystyle \frac{\pi^2}{2}\ln 2$$

16.

$$\int_{0}^{\pi/2}\sin x\ln\sin x dx=\ln 2-1$$

17.

$$\int_{0}^{2\pi}\ln(a+b\sin x)dx=\int_{0}^{2\pi}\ln(a+b\cos x)dx=2\pi\ln(a+\displaystyle \sqrt{a^2-b^2})$$

18.

$$\int_{0}^{\pi}\ln(a+b\cos x)dx=\pi\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-b^2}}{2}\right)$$

19.

$$\int_{0}^{\pi}\ln(a^2-2ab\cos x+b^2)dx=\left\{ \begin{array}{ll} 2... ...space{.3in} a\geq b>0\\ 2\pi\ln b,&\hspace{.3in} b\geq a>0 \end{array}\right.$$

20.

$$\int_{0}^{\pi/4}\ln(1+\tan x)dx=\displaystyle \frac{\pi}{8}\ln2$$

21.

$$\int_{0}^{\pi/2}\sec x\ln\left(\displaystyle \frac{1+b\cos x}{1+a\... ...}\right)dx=\displaystyle \frac{1}{2} \Big[ (\cos^{-1}a)^2-(\cos^{-1}b)^2 \Big]$$

22.

$$\int_{0}^{a}\ln\left(2\sin\displaystyle \frac{x}{2}\right)dx=-\lef... ...e \frac{\sin 2a}{2^2}+\displaystyle \frac{\sin 3a}{3^2}+\cdot\cdot\cdot \right)$$

23.

$$\int_{0}^{a}\ln\left(2\sin\displaystyle \frac{x}{2}\right)dx=-\lef... ...e \frac{\sin 2a}{2^2}+\displaystyle \frac{\sin 3a}{3^2}+\cdot\cdot\cdot \right)$$

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