# Tagged Questions

Questions about the evaluation of specific definite integrals.

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### Solving an equation involving an integral: $\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$

Determine a pair of number $a$ and $b$ for which $$\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$$ I tried putting $x$ as $1-x$ as the integral wouldn't change but could not move forward from ...
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### As I can find the critical points of the function $\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$ [on hold]

Hi I'm stuck with a problem, greatly appreciate a suggestion to solve: As I can find the critical points of the function $$\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$$
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### Is this integral equals zero? [duplicate]

I want to calculate this integral $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ How do I calculate this integral?
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### What is relation between these integrals

I know $$\int_{0}^{\frac{\pi}{2}}\ln(\sin x)dx=-\frac{\pi}{2}\ln(2)$$ What is relation between it and $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ Please guid me. I have sixteen ...
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### On the vanishing of integrals involving the $\sinh$ function. [on hold]

Suppose for some positive real $\theta$ that $$\int_1^\infty f(x)\sinh(\theta\log \sqrt x) \mathrm{d}x = 0$$ Where $f(x)$ is a non-constant and continuous function of $x$. What necessary properties ...
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### Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
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### A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt$

$$\mathbf{\mbox{Evaluate:}}\qquad \int_{0}^{1} \sqrt{\frac{1}{\left(1 - t^{2}\right)^2} - \frac{\left(n + 1\right)^{2}\,t^{2n}}{\left(\, 1 - t^{2n+2}\,\,\right)^{2}}} \,\,\mathrm{d}t$$ where $n$ ...
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### $\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
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### ratio of 2 definite integrals [on hold]

IfA=∫(sin(884x)sin(1122x)/sin(x), x, 0, pi) and B=∫(x^238(x^1768-1)/(x^2-1), x, 0, 1) then the value of A/B? Since in A we can write it as ∫(sin(884x)sin(1122x)/sin(x), x, 0, pi/2)and then x-->π/2-x ...
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### Integral Convergence with parameters

I am finding it hard to approach this question: $$\int_0^{\pi/2} {1-\cos(x)^a\over x^b}\, dx$$ and I need to determine for which positive values of $a,b$ the integral converges. Thanks,
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### Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
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### Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
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### Calculate integral $x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}$ [on hold]

Calculate $\int_0^{1-x2}\ x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}dx_1$ If $n,l,k$ is fixed -it's easy - just expand polynomial and cause similar terms. But what I can do in this common case?
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### Show the triple integral given is equivalent to $\frac{15\pi}{16}$
Evaluate $$\iiint_E\;z \, dV$$ where E is enclosed between the spheres $x^2 + y^2 + z^2 = 1$and$x^2 + y^2 + z^2 = 4$ in the first octant. I'll be honest. My first ...