Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}\frac{x\,\log\tan x}{\sin x}\,dx\approx-0.845708026471324676...$$
2026-04-06 23:04:22.1775516662
Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$
524 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in INTEGRATION
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- How to find the unit tangent vector of a curve in R^3
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- Proving smoothness for a sequence of functions.
- Random variables in integrals, how to analyze?
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Which type of Riemann Sum is the most accurate?
Related Questions in DEFINITE-INTEGRALS
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- Closed form of integration
- Integral of ratio of polynomial
- An inequality involving $\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx $
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Roots of the quadratic eqn
- Area between curves finding pressure
- Hint required : Why is the integral $\int_0^x \frac{\sin(t)}{1+t}\mathrm{d}t$ positive?
- A definite integral of a rational function: How can this be transformed from trivial to obvious by a change in viewpoint?
- Integrate exponential over shifted square root
Related Questions in LOGARITHMS
- Confirmation of Proof: $\forall n \in \mathbb{N}, \ \pi (n) \geqslant \frac{\log n}{2\log 2}$
- Extracting the S from formula
- How to prove the following inequality (log)
- Rewriting $(\log_{11}5)/(\log_{11} 15)$
- How to solve this equation with $x$ to a logarithmic power?
- Show that $\frac{1}{k}-\ln\left(\frac{k+1}{k}\right)$ is bounded by $\frac{1}{k^2}$
- Why do we add 1 to logarithms to get number of digits?
- Is my method correct for to prove $a^{\log_b c} = c^{\log_b a}$?
- How to prove the inequality $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}\geq \log (2)$?
- Unusual Logarithm Problem
Related Questions in CLOSED-FORM
- How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$
- Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$
- Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?
- Finding a closed form for a simple product
- For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?
- Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$
- How can one show that $\int_{0}^{1}{x\ln{x}\ln(1-x^2)\over \sqrt{1-x^2}}\mathrm dx=4-{\pi^2\over 4}-\ln{4}?$
- Exercises about closed form formula of recursive sequence.
- Simplify and determine a closed form for a nested summation
- Direction in closed form of recurrence relation
Related Questions in TRIGONOMETRIC-INTEGRALS
- Sine of the sum of two solutions of $a\cos\theta + b \sin\theta = c$
- Tan of difference of two angles given as sum of sines and cosines
- strange partial integration
- Solution of the trigonometric equation $\sin5x =\cos2x$
- How to compute $\int_{0}^{1}\left (\frac{\arctan x}{1+(x+\frac{1}{x})\arctan x}\right )^2dx$
- What happens to $\int_a^b \sin(\sqrt[n]{x})\,dx$ as $n\to\infty$?
- Prove that $\forall k>0,\ \int_0^{+\infty}\frac{\sin x}{x+k}dx<\frac1k$.
- Calculate Stieltjes Polynomial
- Integral of function from $0$ to $\pi$ using $\sin x=t$ substitution
- Find an appropriate trigonometric substitution of the form x=f(t) to simplify the integral
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?