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Integrals

Short Answer Type

61.

Evaluate the following definite integral:
$integral subscript 0 superscript 2 fraction numerator 5 straight x plus 1 over denominator straight x squared plus 4 end fraction dx$

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62.

Evaluate the following definite integral:
$integral subscript 0 superscript 1 fraction numerator 2 straight x plus 3 over denominator 5 straight x squared plus 1 end fraction dx$

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63.

Evaluate the following definite integral:
$integral subscript 1 superscript 2 left parenthesis log space straight x squared right parenthesis dx.$

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64.

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 2 end superscript space straight x squared space sinx space dx.$

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65.

Evaluate the following definite integral:
$integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space sin space 3 straight x space dx.$

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66.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$

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67.

By using the properties of definite integrals, evaluate the following integral.
$integral subscript 0 superscript straight pi over 2 end superscript space cos squared space dx$

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68.
Prove the following:
$integral subscript 0 superscript straight pi over 2 end superscript space sin cubed straight x space dx space equals space 2 over 3$

Let I = $integral subscript 0 superscript straight pi over 2 end superscript sin cubed straight x space dx$
$equals 1 fourth integral subscript 0 superscript straight pi over 2 end superscript left parenthesis 3 space sinx minus space sin space 3 straight x right parenthesis space dx$
$equals space 1 fourth open square brackets negative 3 space cosx plus 1 third cos space 3 straight x close square brackets subscript 0 superscript straight pi over 2 end superscript$
$equals space 1 fourth open square brackets open parentheses negative 3 space cos straight pi over 2 plus 1 third cos fraction numerator 3 straight pi over denominator 2 end fraction close parentheses minus open parentheses negative 3 space cos 0 plus 1 third cos 0 close parentheses close square brackets equals space 1 fourth open square brackets negative left parenthesis 3 right parenthesis space left parenthesis 0 right parenthesis plus open parentheses 1 half close parentheses left parenthesis 0 right parenthesis plus 3 minus 1 third close square brackets space equals space 1 fourth open parentheses 8 over 3 close parentheses space equals space 2 over 3$