Newbetuts

Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$

Need help with a definite integral $\int_0^{\infty} \frac{x-1}{\sqrt{2^x-1}\ln(2^x-1)}\,dx$ [duplicate]

Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?

Partial fraction expansion for non-rational functions

How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Solution of an integral with strange imprecision of gamma functions

A problem about parametric integral

How to evaluate $\int_0^y\frac{\ln x\ln^2(1-x)}{x}dx$

Calculate Integral using residue theorem

Clarification on the meaning of dx in the integral and differential setting

Computation of $\int_0^{2\pi} \frac{x^2 \sin{x}}{1 - 2a\cos{x} + a^2}$

How do I integrate these functions?

Partial Fraction Decomposition with Arbitrary Constant in $\int\frac{1}{y^4-K^4}dy$.

Elliptic Integrals & Gamma Functions of Rational values.

Integrate $ \int_0^\infty \left( ( x A+I)^{-1} A - \frac{1}{c+x} I \right)\, \mathrm dx $ where $A$ is positive-definite and $c>0$

Asymptotics of an oscillatory integral with a linear oscillator

New posts in integration

A complex approach to integral of cardinal sine [duplicate]

Asymptotic Expansion of an Oscillating Integral

How to prove the closed form of the integral $\int \frac {dx}{\prod_{r=0}^n (x+r)}$

Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$

Need help with a definite integral $\int_0^{\infty} \frac{x-1}{\sqrt{2^x-1}\ln(2^x-1)}\,dx$ [duplicate]

Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?

Partial fraction expansion for non-rational functions

How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Solution of an integral with strange imprecision of gamma functions

A problem about parametric integral

How to evaluate $\int_0^y\frac{\ln x\ln^2(1-x)}{x}dx$

Calculate Integral using residue theorem

Clarification on the meaning of dx in the integral and differential setting

Computation of $\int_0^{2\pi} \frac{x^2 \sin{x}}{1 - 2a\cos{x} + a^2}$

How do I integrate these functions?

Partial Fraction Decomposition with Arbitrary Constant in $\int\frac{1}{y^4-K^4}dy$.

Elliptic Integrals & Gamma Functions of Rational values.

Integrate $ \int_0^\infty \left( ( x A+I)^{-1} A - \frac{1}{c+x} I \right)\, \mathrm dx $ where $A$ is positive-definite and $c>0$

Asymptotics of an oscillatory integral with a linear oscillator