Consider the integrals ${I_1} = \mathop \smallint \limits_0^1 \,{e^{ - x}}{\cos ^2}\,\,x\,dx,\;\,\,{I_2} = \mathop \smallint \limits_0^1 \,{e^{ - {x^2}}}{\cos ^2}\,\,x\,dx,\;\,\,{I_3} = \mathop \smallint \limits_0^1 \,{e^{ - {x^2}}}\;dx$ and ${I_4} = \mathop \smallint \limits_0^1 \,{e^{ - \left( {1/2} \right){x^2}}}\,\,dx$ The greatest of these integral is

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Hard

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Analyzing +4 | -1

Consider the integrals

{I_1} = \int \limits_0^1 \,{e^{ - x}}{\cos ^2}\,\,x\,dx,\;\,\,{I_2} = \int \limits_0^1 \,{e^{ - {x^2}}}{\cos ^2}\,\,x\,dx,\;\,\,{I_3} = \int \limits_0^1 \,{e^{ - {x^2}}}\;dx

and

{I_4} = \int \limits_0^1 \,{e^{ - \left( {1/2} \right){x^2}}}\,\,dx

The greatest of these integral is

View answer and explanation

Correct option: D

{I_4}

Option D (

{I_4}

) is correct because by comparing the integrands over the interval [0,1],

{e^{ - (1/2){x^2}}}

is the greatest, leading to the largest integral value.