# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.

If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle. Hint is to use integration by part but I can't find a good reference for ...
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### I need help with integrating an expression with multiple $x$ and $y$ terms [closed]

The equation is $x^2+y^2 = 3\sqrt{2} x - 5\sqrt{2} y +2xy$. I need to try and find the area under the curve and above the x axis between $x = 0$ and $x = 3\sqrt{2}$. I've heard that implicit ...
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### Defining formulas for first-order linear differential equations.

When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And ...
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### Is this weird function with argument in the integrand continuous? (Fundamental Theorem of Calculus)

Let $f:I \rightarrow \mathbb{R}$ with $I$ interval and $f \in C^\infty(I)$. If $t_0 \in I$, we know by the FTC that $F:I \rightarrow \mathbb{R}$ given by $F(x)=\int_{t_0}^x f(t)dt$ is continuous. But ...
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### Solve an integral by bringing quadratic trinomial to canonical form

The problem is the following: $$\int{\sqrt{4x^2+x}dx}$$ Now once gotten to canonical form of a quadratic trinomial, $ax^2+bx+c=a(x-(\frac{-b}{2x}))^2-\frac{b^2-4ac}{4a}$, the intregral looks like this:...
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### visualization of triple integrals with iterated bounds in different coordinate systems

Lately i've been trying rather hard to write a code in cpp or python that not only calculates the result of a given triple integral, but also depicts the shape of the volume defined by the integral. ...
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### Find $\int\frac{1}{2\sin (x)+3\cos (x)+1}$ $\Tiny{dx}$

Question Evaluate the following integral: $\int\frac{1}{2\sin (x)+3\cos (x)+1} \small{dx}$ Now, I've tried a couple of different substitutions and integrating partially but unfortunately, to no ...
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### Integration by Parts Table Technique With Exponential and Polynomial Higher Order

I come to you with a rather simple question in need of a reference or two from more knowledgable sources. Here was the simple integration by parts problem: However, a rather peculiar table (book-...
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1 vote
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### Definite integral over an infinite product

Evaluate the following integral $$\int_0^\infty\frac{x+1}{x+2}\cdot\frac{x+3}{x+4}\cdot\frac{x+5}{x+6}\cdots dx$$ When I saw this, I was pretty sure that the infinite term must telescope or it must ...
1 vote
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### Centre of Mass in x,y plane

Given an area $R$ in x,y plane, and the density is $\rho(x,y)$ at $(x,y)$, then we can use double integrals to calculate the centre of mass coordinates. Now my question is, intuitively, given any non-...
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### Why don't the bounds in this definite integral change?

The question This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
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### Limits of Riemann sums left and mid endpoint rule.

In my calculus class we have begun talking about integrals. In particular we have begun talking about Reimann sums and how through the limit of a Reimann sum we can integral. But so far all our ...
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### What are the strategies to deal with intractable integrals encountered when solving ODE using variational method.?

I am trying to solve a nonlinear ODE: $$u_{xx}+\tan^2(x)u+gu^2=0$$ using the variational method, and I encountered an intractable integral. What are the strategies to deal with intractable integrals ...
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### Ratio of an increasing monotonic function and its integral is infinity

Inspired by the fact that for functions at the form: $f(x) = \frac{1}{x^\alpha}$ where $\alpha \ge 1$, the intgeral: $\int_{0}^{1} f(x)dx$ diverges to $\infty$ , and the ratio: $\frac{f'(x)}{f(x)}$ ...
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### Uniform convergence and improper integral

I was thinking about this claim (Real analysis question) let ${f_n}$ be a series of continuous functions that converges uniformly to $f(x)=0$ on the interval $[1,\infty)$ and that satisfies the ...
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### Calculate $\int_{a}^{b} e^x \mathrm{d}x$with the aid of the Riemann subtotal.

Calculate $\int_{a}^{b}e^x \mathrm{d}x$ with the aid of the Riemann subtotal. I know the Riemann subtotal and it is defined as follows: $S_n= \sum\limits_{i=1}^{n}{} f(ξ_i)\Delta x_i$. However, I have ...
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### What's the meaning of the integral of the derivative of a distributin function $F$ if $F'(x)$ exists a.e.?
Lemma 2.2 on page 37 of Allan Gut's Probability: A Graduate Course (2nd edition) says : Let F be a distribution function. Then: (a) $F'(x)$ exists a.e., and is non-negative and finite. (b) \$\int_{a}^{...