Integral help?

I was doing a problem in which i used a bit unconventional approach, but in my process, i came across the following integral. but i am not really good in integration, as i am new to the subject, so i want to ask, if the integral is derivable? and if so, how to do this:

$\large{\int\dfrac{sin(\frac{nx}{2}).cos\frac{(n+1)x}{2}}{sin(\frac{x}{2})}.dx}$

#Calculus #Trigonometry #Integration #IndefiniteIntegral #IntegrationTechniques

Easy Math Editor

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Comments

Can you post the original problem??

Sudeep Salgia - 6 years, 4 months ago

the original problem asked for the sum of the series :

$\large{\sum\limits_{r=1}^{n}\frac{1}{r}sin(r\frac{\pi}{3})}$

I was thinking if i could use the identity :

$\large{\sum\limits_{r=1}^{n}cos(rx)=\dfrac{sin(\frac{nx}{2}).cos\frac{(n+1)x}{2}}{sin(\frac{x}{2})}}$

and then integrate this to get the above sum, but as you can see, this was not fruitful.

i shall post this original problem in a new thread. :D

Aritra Jana - 6 years, 4 months ago

$\dfrac{\pi}{3} = x$

$S = \displaystyle \sum_{r=1}^{n}\dfrac{sin(rx)}{r}$

$C = \displaystyle \sum_{r=1}^{n}\dfrac{cos(rx)}{r}$

$C + iS = \displaystyle \sum_{r=1}^{n} \dfrac{e^{irx}}{r}$

$Now~ \displaystyle \sum_{r=1}^{n} x^{r - 1} = \dfrac{1 - x^n}{1 - x}$

$\displaystyle \sum_{r=1}^{n} \dfrac{x^r}{r} = \int \dfrac{1 - x^n}{1 - x}$

Then keeping $e^{ix} = x$ and comparing the imaginary part we can get the answer but I am unable to find this - $\int \dfrac{x^n}{1 - x}$

U Z - 6 years, 4 months ago

@U Z – Any ideas @Jake Lai

U Z - 6 years, 4 months ago

@U Z – @Sudeep Salgia @Ronak Agarwal @Sandeep Bhardwaj @Pranjal Jain @Pratik Shastri @Shashwat Shukla any ideas?

U Z - 6 years, 4 months ago

There was a problem related to this sum.

Krishna Sharma - 6 years, 4 months ago

@Krishna Sharma – really? i wasn't aware of that this actually came in our FIITJEE open test Mains

Aritra Jana - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$