GATE CSE 2018 | Question: 16

Question

The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____

Comments

So whenever there is not mentioned about degree or radians go for radians. So you think when we write pi/4 then where is degree in it, so when we wirte pi we mean 180 degree.

---

$\prod ^{2}$/16 is not degree so we have it to degree.

$\prod $ ->$180^{0}$

$1^{0}$ ->$180^{0}$/$\prod $

$\prod ^{2}$/16 → 45 $\prod $/4  =35.34 degree

now sin(35.34 degree)/2=0.578/2

 

 

 

Answer 1

$\int _{0}^{\frac{\pi}{4} } x \cos (x^{2}) dx$

put x$^{2} = t$

$2x dx = dt $

$t$ will range from  $0$ to $\frac{\pi ^{2} }{16}$

Now our new integral is : $\frac{1}{2}\int _{0}^{\frac{\pi ^{2} }{16}} \cos(t) dt$

= $\frac{1}{2} [\sin (t) ]_{0}^{\frac{\pi ^{2} }{16}} = \frac{1}{2} [\sin (.616225)- 0]$ = $\frac{0.5779}{2}$

 

 0.289 Answer 

Comments

I didnt round it off. I wrote 0.288. Will it be considered correct?

---

why  not  sin(45*45)/2 degreee    =-.353

---

i have typed   .28 will it be correct ???

---

@ sumit goyal 1

plz tell here why we need to change degree to radian ?

---

yeah same question i have

---

Yes Correct

---

@srestha     @rio

main()

{

If ( there is a degree symbol   ∘ )

{

Printf("use degree mode in calculator ") ;

}

else

{

printf("then use radian  mode if degree symbol not given ");

}

}

---

@ sumit goyal 1

Here radian is not mentioned

So, why donot we do it in degree?

---

Can u show me any integration with degree symbol?

---

In stackoverflow someone told that if  degree symbol  is mentioned than go for degree  else go for radian if nothing is mentioned

---

Why i am getting sin value as this ?

---

Here radian is not mentioned

So, why donot we do it in degree?

 The Reason for this question lies in the geometric interpretation of $\lim_{\Theta \rightarrow 0} {}\frac{Sin\Theta }{\Theta } = 1$

In  calculus , $\frac{d}{dx} (sinx)) = cosx$ (or) $\lim_{x\rightarrow 0} {}\frac{sinx}{x} = 1$ is valid  if x is in radian.

So, Differentiation/Integration of all trigonometric and inverse trigonometric functions is valid if x is in radian.

-----------------------------------------------------------------------------------------------------------------

Geometric Interpretation of $\lim_{\Theta \rightarrow 0} {}\frac{Sin\Theta }{\Theta } = 1$ :-

Suppose, We have a circle with radius 'r' = 1 unit

---

i think they have mentioned to take pi=3.14 that is why we have to calculate it in radian if nothing is mentioned then pi=180 is taken and sin value is calculated in degree .

so basically if pi=180 then it is calculated in degree and if pi=3.14 then it will be calculated in radian.

---

@ankitgupta.1729 👍

---

Choose the radian radio button.

---

I find .616 then do .616/2 but my answer not matched where is the mistake? Will anybody suggest?

---

can you tell me if

x^2=t

2xdx=dt

dx=dt/2x

so, why you have written 1/2 ∫cos(t)dt

where is the x in 1/2 ?

---

In place of (x)dx dt/2 is written and in place of cos(x^2) cos(t) is written. Hope It is clear if you have any other doubt please write again.

---

@Sachin Mittal 1 @Deepak Poonia  @gatecse 

Sir here we will get something like 0.288979318.

Sir my question is what does correct to three decimal places means. Will it be just copy as it is till the third decimal place(0.288) or should it be round off till three decimal places(0.289) ?

Answer 2

Let ${x^2}$=t, So, 2xdx=dt

when ${x^2}$=0, t =0 and when ${x^2}$=${\Pi /4}$, t= $\Pi ^{2}/16$

Substituting it in the question, the integral becomes 

$\int_{0}^{\Pi /4}$cos(t)/2 dt =$[sin t/2]_{0}^{\Pi^{2}/16}$

=sin(0.617 radians)-0 / 2

=0.5785909/2

=0.2892 (ANS)

Comments

i have filled .28 will it be count ??

Answer 3

0.2892

Solved by letting x² =t

Comments

@srestha

I'm also stuck with this why conversion is required??

---

U  did mistake in degree and radian concept , calculate it in raadian mode of calc.

---

GATE is all about Trapping

Answer 4

I didnt round it off. I wrote 0.288. Will it be considered correct?

Comments

same here i wrote.28 don't know whether it will count or not