## How do you evaluate this integral $\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$

$$\int \frac{6x^{3}+7x^2-12x+1}{\sqrt{x^2+4x+6}}dx$$

I have met this complex integral today, how do you evaluate this? Can you do this in more steps?

## Where can I find some free benchmarks to evaluate a MCU?

At present, I’m designing a soft processor with single-precision floating point unit (FPU). I am going to put my soft core into an FPGA and do some performance evaluation. The benchmarks are supposed to contain many single-precision floating point calculations, such as fadd, fmul, fdiv and fsqrt. Therefore, I can know the speed-up provided by FPU. Besides, I also want to compare my soft core with popular cores from ARM such as Cortex-M0, M1, M3. Since ARM provides DesignStart of these cores, it should be easy to put them into an FPGA and do comparision.

I only know 2 free benchmarks that could run on a soft core: Dhrystone and CoreMark. Can anyone share other free benchmarks for evaluation? Spending 1000+ dollars to buy a benchmark is not affordable for me.

## Evaluate double integral bounded by lines and hyperbolas

Evaluate the integral $$\iint_R x^2 y^2 dx dy,$$ where $$R$$ is the bounded portion of the first quadrant bounded by the lines $$y=x, y=4x$$ and the hyperbolas $$xy=1$$ and $$xy=2.$$

Based on the graph of this region, I could split the double integral into three parts based on the range of $$x$$ values ($$1/2$$ to $$\sqrt 2/2,$$ $$\sqrt 2/2$$ to $$1$$, and $$1$$ to $$\sqrt 2$$). This is feasible, but I’m wondering if there’s a simpler way?

## What technique(s) would you use to evaluate and [re]define fields in a lengthy and complex form?

We were commissioned the redesign of a web application for real estate agencies. During the first stage of our user research, which was a contextual inquiry, we found that the form fields (or data structure) for entering a new Property has many problems. There are fields that should not be mandatory, malformed fields, fields with incorrect options, maybe missing fields, etc.

We are looking for a user-research technique or method that allows us to evaluate the existing fields and determine the necessary changes in order to adapt the forms to current market practices and customer needs. The usual methods such as card sorting would not be very useful here, as it’s not a matter of labeling or sorting the fields.

Since we have seen differences of opinion between the real estate agents and some level of consensus must be achieved, I suspect we should use a quantitative technique rather than a qualitative one, but we are open to all possibilities.

Any ideas?

(Context: This is an application for the housing market in a latin american country with thousands of real estate agents, closed to the general public, which has been running for more than a decade with little to no UI or UX updates).

by: AalfaSEO
Created: —
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## How I could evaluate this integral?

$$I(n)=\int_0^1 \frac{x^n}{x^2+x+1} \,dx$$ I’ve tried so far integration by parts and partial fraction, each method has led to a dead end. So how I should evaluate this using standard methods, if is possible of course.

## How to evaluate $\int_{S^2} \sin(\theta) d\theta \wedge d\phi$?

I’m kind of stuck here. I try to apply Stokes Theorem, so that $$\int_{S^2} \sin(\theta) d\theta \wedge d\phi = \int_{D^2} d(\sin(\theta) d\theta \wedge d\phi) = 0$$ which is not the answer?

## can’t evaluate integral of the form $\log \left(2+\lambda _+ P+ 2 \sqrt{\lambda _* P^2+\lambda _+ P+1}\right)$ times a fraction

I’m trying to evaluate a bunch of integrals of the form $$\int_0^{\infty } \frac{\sum\limits_{i=0}^k (c_i P^i)}{\left(P^2+1\right)^n}\log \left(2+\lambda _+ P+ 2 \sqrt{\lambda _* P^2+\lambda _+ P+1}\right)\, dP$$ where $$n$$ and $$k$$ is are integers ($$n\ge 1$$, $$m<2n$$), $$\lambda_+>0$$, $$\lambda_*>0$$, and the polynomial $$\sum\limits_{i=0}^m(c_i P^i)$$ only has even powers of $$P$$; for example $$\frac{3-2P^2 +47P^4+4P^6}{(1+P^2)^5}$$.

For each particular term of the form $$\int_0^{\infty } \frac{1}{\left(P^2+1\right)^k}\log \left(2+\lambda _+ P+ 2 \sqrt{\lambda _* P^2+\lambda _+ P+1}\right)\, dP$$ parital integration $$\int u \,dv = u v – \int v\,du$$ (with $$dv=log(\ldots)$$) will give me a polynomial of the order $$2n-4$$ divided by $$1/(1+P^2)^{n-1}$$ (which can easily be solved) plus $$\int\frac{\arctan(P)}{P}\left(1-\frac{1}{\sqrt{\lambda _* P^2+\lambda _+ P+1}}\right)dP$$ which I don’t know how to solve (nor does Mathematica apparently). In certain instances (for particular set of $$c_i$$ the (indefinite) integral can be solved by partial integration when the $$\arctan(x)$$ terms cancel out (for example $$\frac{1-16P^2 +25P^4-6P^6}{(1+P^2)^5}$$).

The queation is how do I solve the integral when $$\arctan(x)$$ terms don’t cancel out? I have tried all the variable substitutions I could think of, and couldn’t figure anything out by trying the residue theorem either.

Any help would be greatly appreciated.

## Can someone please evaluate this indefinite integral for me! It is [integral] e^x / sqrt(e^x + e^(2x) dx

I started out with U-sub, making e^x as U. Replaced everything with U and now I am stuck!

## How can I evaluate the safety of these Magsafe 2 USB-C Charge Cables?

They look like and claim to be USB-C to Magsafe 2 cables, that allow you to charge up on the go from a USB-C wall adaptor.