GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
64 views
Let $\frac{\mathrm{d} }{\mathrm{d} x}f(x)$ = $\frac{e^{sinx}}{x}, x>0$ if $\int_{1}^{4}\frac{2e^{sinx^{2}}}{x}d(x)$ = f(k)-f(1) then k = ______
asked in Calculus by Boss (7.3k points)  
reshown by | 64 views

1 Answer

+2 votes
Best answer

f(k) - f(1)

= $\int_{-\infty }^{k}\frac{e^{sinx}}{x}$ - $\int_{-\infty}^{1}\frac{e^{sinx}}{x}$

= $\int_{1}^{k}\frac{e^{sinx}}{x}$    ...................(1)

 

Now, we simplify the integral

I = $\int_{1}^{4}\frac{2e^{sinx^{2}}}{x}$

 

Put x2 = t

$\therefore 2x dx=dt$

$\therefore dx=\frac{dt}{2\sqrt{t}}$

 

$\therefore$ I = $\int_{1}^{16} \frac{2e^{sint}dt}{\sqrt{t}*2\sqrt{t}}$   ...........(2)

 

Now, when we simplify statement (2), we get statement (1).

 

$\therefore k=16$

answered by Veteran (14.9k points)  
selected by
Two errors:

1. Statement 2 should be integration from 1 to 16.

2. Statement 2 should be integration of $\frac{2e^{sint}}{\sqrt{t}*2\sqrt{t}}$
Thanks . will edit

Related questions

+1 vote
2 answers
1
asked in Calculus by mcjoshi Veteran (24.6k points)   | 204 views


Top Users Apr 2017
  1. akash.dinkar12

    3366 Points

  2. Divya Bharti

    2536 Points

  3. Deepthi_ts

    2040 Points

  4. rude

    1966 Points

  5. Tesla!

    1768 Points

  6. Shubham Sharma 2

    1610 Points

  7. Debashish Deka

    1584 Points

  8. Prashant.

    1462 Points

  9. Arunav Khare

    1444 Points

  10. Kapil

    1414 Points

Monthly Topper: Rs. 500 gift card

22,072 questions
28,030 answers
63,194 comments
24,128 users