2
votes
2answers
31 views

Integral expression $P=x^3+x^2+ax+1$, …

I am trying to solve the following problem: Consider the integral expression $P=x^3+x^2+ax+1$, where $a$ is a rational number. At $a = ?$ the value of $P$ is a rational number for any x which ...
3
votes
5answers
192 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
1
vote
1answer
32 views

consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.

Any ideas how to start. I am having trouble figuring out the region itself All ideas are appreciated thanks
-1
votes
1answer
66 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
2
votes
0answers
84 views

Calculate the Gauss integral without first squaring it

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a 2-d integral in the plane and integrate it in polar ...
1
vote
2answers
47 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
3
votes
3answers
78 views

If $\int_{-\infty}^{\infty}f(x)\ \mathrm dx=100$ then $\int_{-\infty}^{\infty}f(100x+9)\ \mathrm dx =?$

Given $\displaystyle\int_{-\infty}^{\infty}f(x)\ dx=100$, evaluate $\displaystyle\int_{-\infty}^{\infty}f(100x+9)\ dx.$ Question is as above. I'm not sure how to even start. Is the answer $100$? ...
0
votes
1answer
50 views

Solve $f(x) = x^n$ for $n$ given the value of the definite integral of $f(x)$ between some values $a$ and $b$

I have a curve $f(x) = x^n$, where $n$ is variable. The curve will be plotted from the $f(a)$ to $f(b)$, where $a$ and $b$ are positive integers, and $b > a$. I want to find $n$ given the desired ...
2
votes
3answers
81 views

Laplace transform of the following function

find the laplace transform of the function : $$f(t) =\begin{cases} t^2, & 0<t<1 \\ 2\cos t+2, & t>1 \\ \end{cases}$$ My attempt: $$L\{f(t)\}=\int_{0}^{1}e^{-st} \ t^2 \ ...
0
votes
1answer
52 views

Solve the following integral

How $$\frac{1}{\pi}\left[\int_{-\pi}^{-\frac{\pi}{2}}\sin2x\cos nxdx+\int_{0}^{\pi}\sin2x\cos nxdx\right]=\frac{-2}{\pi}\frac{1+\cos(\frac{n\pi}{2})}{n^2-4}$$ $n=1,2,3,\ldots$ ?? My Attempt ...
0
votes
2answers
20 views

Polynomial whose n no. of integrals are zero

Is it true that, say $y(x)$ be a polynomial of degree $\le$ n, such that $\int^1_0 x^i y(x) dx = 0$ for $0 \le i \le n$ then y(x) is zero polynomial ? Prove or disprove.
0
votes
1answer
45 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
0
votes
1answer
44 views

Please help finishing the calculation to find the Entropy of Pareto distribution.

Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, ...
1
vote
0answers
61 views

Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.

Let X follows Pareto distribution with parameters α, a, h. that is X~Pa(α,a,h) Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter. ...
2
votes
2answers
2k views

change of variables for definite integrals

First of all I would like to start off by asking why do they have different change of variable formulas for definite integrals than indefinite...why cant we just integrate using U substitution as we ...
3
votes
2answers
261 views

Integrating $\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x$

I've been trying to integrate this: $$\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x .$$ Unfortunately I haven't found a way so far. I've been trying to factor the denominator in order to end up ...