Let I=\int_{0}^{1}\frac{\sin x}{\sqrt{x}}dx\; and\; J=\int_{0}^{1}\frac{\cos x}{\sqrt{x}}dx. Then which one of the following is true?

  • Option 1)

    I> \frac{2}{3}\; and\; J< 2

  • Option 2)

    I> \frac{2}{3}\; and\; J> 2

  • Option 3)

    I< \frac{2}{3}\; and\; J< 2

  • Option 4)

    I< \frac{2}{3}\; and\; J> 2

 

Answers (1)

As we learnt in

Properties of Definite Integration -

If f(x)\geqslant g(x) for

all x\in \left [ a,b \right ] then

\int_{a}^{b}f\left ( x \right )dx\geqslant \int_{a}^{b}g(x)dx

-

 

 I_{1}=\int_{0}^{1}\frac{sinx}{\sqrt{x}}dx<\int_{0}^{1}\frac{x}{\sqrt{x}}dx=\int_{0}^{1}\sqrt{x}dx

So, I_{1}<\frac{2}{3}

Now, J = \int_{0}^{1}\frac{cosx}{\sqrt{x}}dx\leq \int_{0}^{1}\frac{1}{\sqrt{x}}dx=2

J\leq 2

 


Option 1)

I> \frac{2}{3}\; and\; J< 2

Incorrect

Option 2)

I> \frac{2}{3}\; and\; J> 2

Incorrect

Option 3)

I< \frac{2}{3}\; and\; J< 2

Correct

Option 4)

I< \frac{2}{3}\; and\; J> 2

Incorrect

Exams
Articles
Questions