Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
Questions tagged [trigonometric-integrals]
1789 questions
60
votes
9 answers
How to evaluate $\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$
Find the value of
$$I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$$
We have the information that
$$J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)\ \mathrm dx=\dfrac{\pi^2}{8}\ln^2(2)-\dfrac{\pi^4}{192}$$
math110
- 94,932
- 17
- 148
- 519
41
votes
16 answers
Is there another way to solve $\int_{0}^{\pi}\frac{1-\sin x}{\sin x+1}dx$
My way to solve this integral. I wonder is there another way to solve it as it's very long for me.
$$\int_{0}^{\pi}\frac{1-\sin (x)}{\sin (x)+1}dx$$
Let $$u=\tan (\frac{x}{2})$$
$$du=\frac{1}{2}\sec ^2(\frac{x}{2})dx $$
By Weierstrass…
Mathxx
- 8,468
37
votes
5 answers
Compute close-form of $\int_0^{\frac\pi2}\frac{dt}{\sin t+\cos t+\tan t+\cot t+\csc t+\sec t}$
I came across the improper trigonometric integral recently shared in an on-line forum
\begin{align}
\int_0^{\frac\pi2}\frac{dt}{\sin t+\cos t+\tan t+\cot t+\csc t+\sec t}\\
\end{align}
which amuses me because of its appearance. What is more amusing…
Quanto
- 120,125
36
votes
4 answers
Bizarre Integral $\int_0^1 \frac{\tan^{-1}{\frac{88\sqrt{21}}{215+36x^2}}}{\sqrt{1-x^2}} {d}x = \frac{\pi^2}{6}$
Does the following equality hold?
$$\int_0^1 \frac{\tan^{-1}{\frac{88\sqrt{21}}{215+36x^2}}}{\sqrt{1-x^2}} \, \text{d}x = \frac{\pi^2}{6}$$
The supposed equality holds to 61 decimal places in Mathematica, which fails to numerically evaluate it after…
Jack Tiger Lam
- 4,674
31
votes
10 answers
Various ways to calculate $\int \sin(x) \cos(x) \, \mathrm{d}x$
Consider the integral
$$\mathcal{I} := \int \sin(x) \cos(x) \, \mathrm{d} x$$
$
\newcommand{\II}{\mathcal{I}}
\newcommand{\d}{\mathrm{d}}
$
This is one of my favorite basic integrals to think about as an instructor, because on the face of it, there…
PrincessEev
- 50,606
30
votes
1 answer
Evaluating $\left.\int_0^{\pi/2}\sqrt{1+\frac1{\sqrt{1+\tan^nx}}}\text dx \middle/\int_0^{\pi/2}\sqrt{1-\frac1{\sqrt{1+\tan^nx}}}\text dx\right.$
Evaluate the integral ratio$$\dfrac{I_1}{I_2}=\dfrac{\displaystyle \int_{0}^{\frac{\pi}{2}} \sqrt{1+\frac{1}{\sqrt{1+\tan^nx}}} \mathrm{d}x}{\displaystyle \int_{0}^{\frac{\pi}{2}} \sqrt{1-\frac{1}{\sqrt{1+\tan^nx}}} \mathrm{d}x}$$
Using…
V.G
- 4,262
30
votes
10 answers
Why $ \int_0^{2\pi}\frac{\cos t - r}{1 - 2r\cos t + r^2}\,dt=0$
Define
$$
I(r) = \int_0^{2\pi}\frac{\cos t- r}{1 - 2r\cos t + r^2}\,dt
$$
Numerical experiments suggest that $I(r) = 0$ for all $r\in [0,1)$. But I can't show this analytically.
This integral appears when computing the Cauchy transform of $\overline…
amsmath
- 10,049
29
votes
4 answers
Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$
Please help me to evaluate this integral:
$$\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$$
Using substitution $x=2\arctan t$ it can be transformed to:
$$\int_0^\infty\frac{2}{1+t^2}\arctan^2\left(\frac{2t}{3+t^2}\right)dt$$
Then I tried…
29
votes
7 answers
How to evaluate $I=\int\limits_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$
Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int\limits_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$
The numerical value is I=-1.122690024730644497584272...
How to evaluate this integral?
By against,I find:
…
user178256
- 5,783
28
votes
1 answer
Prove $\int_0^{\sqrt2/4}\frac1{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)(x-1+x\sqrt{9-16x})}{1-2x}}dx=\frac{\pi^2}{8}$ (from a probability question)
Let $$I=\int_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)(x-1+x\sqrt{9-16x})}{1-2x}}dx$$
Prove that $I=\dfrac{\pi^2}{8}$.
Wolfram suggests that it's true but does not find the antiderivative.
Here is the graph of the function being…
Dan
- 35,053
28
votes
6 answers
An interesting trigonometric integral
Yesterday I took part in an entry competition to one of the MSc programs in my university, and during one of the mathematical tests, I had to solve some integral and differential equations.
One of those integrals had the following trigonometric…
o.spectrum
- 1,376
- 11
- 21
27
votes
2 answers
Show that $\int_0^{\pi/3}\arccos^2(2\sin^2 x-\cos x)\mathrm dx=\frac{19\pi^3}{135}$
Show that $$\int_0^{\pi/3}\arccos^2(2\sin^2 x-\cos x)\mathrm dx=\frac{19\pi^3}{135}$$
Wolfram strongly suggests that it's true.
Here is the graph of $y=\arccos^2(2\sin^2 x-\cos x)$.
Context
I stumbled upon this when I was trying to answer a…
Dan
- 35,053
26
votes
6 answers
Help with $\int _0^{\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x$
I want to know how to prove that
$$\int _0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=\frac{4G}{\pi }$$
Here $G$ denotes Catalan's constant, I obtained such result with the help of mathematica.
I also found that the…
Tamayo
- 250
26
votes
4 answers
Question about finite analog of $\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8}$
The integral
$$
\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8},
$$
is given as equation $(17)$ in M.L. Glasser, Some integrals of the Dedekind $\eta$-function.
More general integral
$$
\int_0^\infty…
Cave Johnson
- 4,175
26
votes
4 answers
Twist on log of sine and cosine integral $\int_{0}^{\frac{\pi}{2}}x\ln(\sin x)\ln(\cos x)dx$
I ran across this integral and have not been able to evaluate it.
$$\displaystyle \int_{0}^{\frac{\pi}{2}}x\ln(\sin(x))\ln(\cos(x))dx=\frac{{\pi}^{2}\ln^{2}(2)}{8}-\frac{{\pi}^{4}}{192}$$
I had some ideas. Perhaps some how arrive at…
Cody
- 14,440