Question

$Let space straight I space space equals space integral subscript 0 superscript 1 space fraction numerator sin space straight x over denominator square root of straight x end fraction dx space and space straight J space equals space integral subscript 0 superscript 1 space fraction numerator cos space straight x over denominator square root of straight x end fraction space dx$ then which one is true?

I > 2/ 3 and J > 2

I < 2/ 3 and J < 2

I < 2/ 3 and J > 2

I > 2/ 3 and J < 2

Solution

I < 2/ 3 and J < 2

straight I equals space integral subscript 0 superscript 1 fraction numerator sin space straight x over denominator square root of straight x end fraction space dx space less than space integral subscript 0 superscript 1 space fraction numerator straight x over denominator square root of straight x end fraction dx space equals space integral subscript 0 superscript 1 square root of straight x dx space equals space 2 over 3 straight x to the power of 3 divided by 2 end exponent space left enclose space subscript 0 superscript 1 end enclose space equals space 2 over 3 rightwards double arrow straight l space less than space 2 over 3 straight J space equals space integral subscript 0 superscript 1 fraction numerator cos space straight x over denominator square root of straight x end fraction space dx space less than integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of straight x end fraction space dx space equals space 2 square root of straight x vertical line subscript 0 superscript 1 space equals space 2 therefore space straight J less or equal than 2

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