Let I1=∫ limits0π / 2 e-x2 sin (x) d x ; I2=∫ limits0π / 2 e-x2 d x ; I3=∫ limits0π / 2 e-x2(1+x) d x and consider the statements text I I 1

Question

Let I1=∫ limits0π / 2 e-x2 sin (x) d x ; I2=∫ limits0π / 2 e-x2 d x ; I3=∫ limits0π / 2 e-x2(1+x) d x and consider the statements text I I 1< I 2 text II I 2< I 3 text III I 1= I 3 Which of the following is(are) true? (A) I only (B) II only (C) Neither I nor II nor III (D) Both I and II

Tardigrade Admin · Accepted Answer

since 0< sin x< 1 and 1+x >1 in (0, π / 2) hence I 3> I 2> I 1 ⇒ text A and B are correct ⇒ text (D)

Q. Let $I_{1} = 0 \int π /2 e^{- x^{2}} sin (x) d x; I_{2} = 0 \int π /2 e^{- x^{2}} d x; I_{3} = 0 \int π /2 e^{- x^{2}} (1 + x) d x$ and consider the statements
$I I_{1} < I_{2}$
$II I_{2} < I_{3}$
$III I_{1} = I_{3}$
Which of the following is(are) true?

I only

II only

Neither I nor II nor III

Both I and II

since $0 < sin x < 1$ and $1 + x > 1$ in $(0, π /2)$
hence $I_{3} > I_{2} > I_{1}$
$\Rightarrow A and B are correct \Rightarrow (D)$

Q. Let I1​=0∫π/2​e−x2sin(x)dx;I2​=0∫π/2​e−x2dx;I3​=0∫π/2​e−x2(1+x)dx and consider the statements I I1​<I2​ II I2​<I3​ III I1​=I3​ Which of the following is(are) true?