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Integrals

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Question

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

B.

I < 2/ 3 and J < 2

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Question

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$let space straight F left parenthesis straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis space plus space straight f space open parentheses 1 over straight x close parentheses comma space where space straight f left parenthesis straight x right parenthesis space equals space integral subscript 1 superscript straight x fraction numerator log space straight t over denominator 1 plus straight t end fraction dt.$ Then F(e) equals

1/2

1

2

0

Solution

A.

1/2

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Question

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$integral subscript 0 superscript straight pi space xf space left parenthesis sin space straight x space right parenthesis space dx$ is equal to
$straight pi integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx$
$straight pi integral subscript 0 superscript straight pi space straight f left parenthesis sin space straight x right parenthesis space dx$
$straight pi over 2 integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx$
$straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f left parenthesis cos space straight x right parenthesis space dx$

Solution

D.

straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f left parenthesis cos space straight x right parenthesis space dx

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Question

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$integral subscript negative 3 straight pi divided by 2 end subscript superscript negative straight pi divided by 2 end superscript open square brackets open parentheses straight x plus straight pi close parentheses cubed plus space cos space left parenthesis straight x space plus 3 straight pi right parenthesis close square brackets dx space is space equal space to$
$straight pi to the power of 4 over 32$
$straight pi to the power of 4 over 32 space plus space pi over 2$
$pi over 2$
$pi over 4 minus 1$

Solution

C.

pi over 2

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Question

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$integral space open curly brackets fraction numerator left parenthesis log space straight x minus 1 right parenthesis over denominator 1 plus left parenthesis log space straight x right parenthesis squared end fraction close curly brackets squared space dx space is space equal space to$
$fraction numerator log space straight x over denominator left parenthesis log space straight x right parenthesis squared plus 1 end fraction plus space straight C$
$fraction numerator straight x over denominator straight x plus 1 end fraction space plus straight C$
$fraction numerator xe to the power of straight x over denominator 1 plus straight x squared end fraction plus straight C$
$fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C$

Solution

D.

fraction numerator straight x over denominator left parenthesis log space straight x right parenthesis squared space plus space 1 end fraction plus space straight C

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