I need to prove that this integral diverges:
$$\int_{-\infty}^{\infty}\frac{\cos{x}}{(x^2-3x-4)x}dx$$
And I don't know why. Any help? Thanks!
I know that there are 3 real singularities:
$x=0$,
$x=-1$,
$x=4$
2026-03-31 23:26:43.1774999603
Divergence of improper integral
64 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related 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 IMPROPER-INTEGRALS
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- prove that $\int_{-\infty}^{\infty} \frac{x^4}{1+x^8} dx= \frac{\pi}{\sqrt 2} \sin \frac{\pi}{8}$
- Generalized Fresnel Integration: $\int_{0}^ {\infty } \sin(x^n) dx $ and $\int_{0}^ {\infty } \cos(x^n) dx $
- Need a guide how to solve Trapezoidal rule with integrals
- For which values $p$ does $\int_0^\infty x\sin(x^p) dx $ converge?
- Proving $\int_0^1\frac{dx}{[ax+b(1-x)]^2}=\frac1{ab}$
- Contour integration with absolute value
- Use the comparison test to determine whether the integral is convergent or divergent.
- Can I simply integrate this function?
Related Questions in DIVERGENT-SERIES
- Proving that a series is divergent?
- Is this : $\sum_{n=1}^{+\infty}\frac{(-1)^n}{\tan(n!)}$ convergent sum?
- Convergence (or Divergence) of [(-1)^(n-1)*e^(1/n)]/n
- Show that $\sum_{n=1}^\infty \frac{1}{(\log(n))^p}$ diverges
- Showing Harmonic series is divergent. (question on summation properties)
- Testing convergence of $\sum\limits_{n=1}^{\infty}u_n$ , where $u_n = \left ( 4- \frac{1}{n}\right) ^ { \frac{( - 1) ^ {n }}{ n}}$
- how to prove $\sum \frac {|\alpha+\sin(n^2)|}n$ diverges without summation by parts?
- Behavior of a sum on the boundary of convergence/divergence
- Divergence of a Series $\sum_{n=1}^\infty (\frac{1}{n!})(\frac{n}{e})^n$
- $\sum_{n=1}^{\infty} n $ equals to another value than $\frac1{12}$
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?
Hint. Just because for $x$ near $0$, we have $$ \frac{\cos{x}}{(x^2-3x-4)x} \sim -\frac{1}{4x} \tag1 $$ then, for $\epsilon, \,b $ near $0^+$, we have $$ \int_{\large \epsilon}^b \frac{\cos{x}}{(x^2-3x-4)x} dx \sim -\frac{1}{4}\int_{\large \epsilon}^b\frac{1}{x}dx=-\frac{1}{4}(\log(b)-\log(\epsilon)) \tag2 $$ and, as $\epsilon \to 0^+$, the right hand side of $(2)$ is divergent with your initial integral.