I want to find an expression for

My initial thought was to use the known

Alan Smith
2021-12-26
Answered

I want to find an expression for

My initial thought was to use the known

You can still ask an expert for help

zesponderyd

Answered 2021-12-27
Author has **42** answers

Let $f\left(x\right)=\mathrm{arctan}\left(\frac{x+1}{x-1}\right),\text{}\text{}\mathrm{\forall}\left|x\right|1$

It can be easily showed that:

$f}^{\prime}\left(x\right)=\frac{1}{1+{x}^{2}}=\sum _{n=0}^{\mathrm{\infty}}{(-1)}^{n}{x}^{2n$

Integrating both sides yields that$\mathrm{\exists}C\in \mathbb{R}$ such that:

$\mathrm{arctan}\left(\frac{1+x}{1-x}\right)+C=\sum _{n=0}^{\mathrm{\infty}}\frac{{(-1)}^{n}{x}^{2n+1}}{2n+1}$

Check for f(0) to conclude C and you're done.

It can be easily showed that:

Integrating both sides yields that

Check for f(0) to conclude C and you're done.

Annie Levasseur

Answered 2021-12-28
Author has **30** answers

Use the identity $\mathrm{arctan}\left(\frac{x+1}{x-1}\right)=-\mathrm{arctan}\left(\frac{x+1}{1-x}\right)=-(\frac{\pi}{4}+\mathrm{arctan}\left(x\right))$

karton

Answered 2022-01-08
Author has **368** answers

Since

you can express the RHS as a power series, and then integrate the result to get the desired series for your original function

asked 2020-10-18

If $\mathrm{sin}x+\mathrm{sin}y=a{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{cos}x+\mathrm{cos}y=b$ then find $\mathrm{tan}(x-\frac{y}{2})$

asked 2021-06-16

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 11 passengers per minute.

asked 2021-12-29

I am having trouble finding the correct answer to this question:

Find the angles between$-{360}^{\circ}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{180}^{\circ}$ such that $\sqrt{2}\mathrm{sin}({90}^{\circ}-x)+1=0$

Find the angles between

asked 2021-11-08

For Exercises 56-58, use $\mathrm{log}}_{b}2\approx 0.289,{\mathrm{log}}_{b}3\approx 0.458,\text{}\text{and}\text{}{\mathrm{log}}_{b}5\approx 0.671\text{}\text{to approximate the value of the given logarithms.$

$56.{\mathrm{log}}_{b}8$

$57.{\mathrm{log}}_{b}45$

$58.{\mathrm{log}}_{b}\left(\frac{1}{9}\right)$

asked 2021-06-05

asked 2021-02-21

asked 2021-12-30

Proving $\frac{2\mathrm{sin}x+\mathrm{sin}2x}{2\mathrm{sin}x-\mathrm{sin}2x}={\mathrm{csc}}^{2}x+2\mathrm{csc}x\mathrm{cot}x+{\mathrm{cot}}^{2}x$

Proving right hand side to left hand side:

$\mathrm{csc}}^{2}x+2\mathrm{csc}x\mathrm{cot}x+{\mathrm{cot}}^{2}x=\frac{1}{{\mathrm{sin}}^{2}x}+\frac{2\mathrm{cos}x}{{\mathrm{sin}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x$ (1)

$=\frac{{\mathrm{cos}}^{2}x+2\mathrm{cos}x+1}{{\mathrm{sin}}^{2}x}$ (2)

$=\frac{1-{\mathrm{sin}}^{2}x+1+\frac{\mathrm{sin}2x}{\mathrm{sin}x}}{{\mathrm{sin}}^{2}x}$ (3)

$=\frac{\frac{2\mathrm{sin}x-{\mathrm{sin}}^{3}x+\mathrm{sin}2x}{\mathrm{sin}x}}{{\mathrm{sin}}^{2}x}$ (4)

$=\frac{2\mathrm{sin}x-{\mathrm{sin}}^{3}x+\mathrm{sin}2x}{{\mathrm{sin}}^{3}x}$ (5)

I could not prove further to the left hand side from here.

Proving right hand side to left hand side:

I could not prove further to the left hand side from here.