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Calculus Level 4

$\large \int_{-\frac {\pi}{4}}^{n\pi - \frac {\pi}{4}} \mid \sin{x} + \cos{x} \mid \ \mathrm{d}x$

For all natural numbers $n$ , find a closed form for the above integral.

2\sqrt{2}/n

2/n

2\sqrt{3}/n

3n

2\sqrt{2}n

2\sqrt{5}/n

2n

3/n

1 solution

Deepanshu Gupta
Mar 7, 2015

too easy !
$\displaystyle{I=\int _{ \cfrac { -\pi }{ 4 } }^{ n\pi -\cfrac { \pi }{ 4 } }{ \left| \sqrt { 2 } \sin { (x+\cfrac { \pi }{ 4 } ) } \right| } dx\\ I=\sqrt { 2 } \int _{ 0 }^{ n\pi }{ \left| \sin { t } \right| } dt=\sqrt { 2 } n\int _{ 0 }^{ \pi }{ \left| \sin { t } \right| dt } \\ I=2\sqrt { 2 } n}$

JEE style : Put $n=1$ , and ans comes out to be $2 \sqrt{2}$ . Only one option follows it. :P

Nice solution. Keep it up. @Deepanshu Gupta

Sandeep Bhardwaj - 6 years, 3 months ago

cool Sir ! $\ddot\smile$ I was also thinking initially that I have to put n=1 , But there are too many options so i think that may be i get multiple option match. So Ieave that . But next I will check all option first . :)

Deepanshu Gupta - 6 years, 3 months ago

You are right. 2 options would have become same if you did that.

Rajdeep Dhingra - 6 years, 3 months ago

I will remember that.

Rajdeep Dhingra - 6 years, 3 months ago

did the same."JEE style".

mudit bansal - 6 years, 3 months ago

You can try this .It is better

Rajdeep Dhingra - 6 years, 3 months ago

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