Definite Integration Question 281
Question: Let $ \frac{d}{dx}F(x)=( \frac{{e^{\sin x}}}{x} );x>0 $ . If $ \int _{1}^{4}{\frac{3}{x}{e^{\sin x^{3}}}dx=F(k)-F(1)} $ , then one of the possible value of k, is
[AIEEE 2003]
Options:
A) 15
B) 16
C) 63
D) 64
Show Answer
Answer:
Correct Answer: D
Solution:
$ \frac{d}{dx}F(x)=\frac{{e^{\sin x}}}{x} $
$ \Rightarrow \int _{1}^{4}{\frac{3}{x}{e^{\sin x^{3}}}dx=\int _{1}^{4}{\frac{3x^{2}}{x^{3}}{e^{\sin x^{3}}}dx}} $
Put $ x^{3}=t\Rightarrow 3x^{2}dx=dt $
$ F(t)=\int_1^{64}{\frac{{e^{\sin t}}}{t}}dt=\int_1^{64}{F(t)dt=F(64)-F(1)}, $
On comparing, $ k=64. $
BETA
