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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:
$integral subscript 0 superscript straight pi over 2 end superscript space sin cubed straight x space dx space equals space 2 over 3$

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69.

Evaluate the following
$integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 straight x space dx$

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70.
Evaluate the following:
$integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 space straight x space dx$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 straight x space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript space tan squared straight x. space tan squared straight x space dx$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space tan squared straight x left parenthesis sec squared straight x minus 1 right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis tan squared straight x space sec squared straight x minus tan squared straight x right parenthesis space dx$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space left parenthesis tan squared straight x space sec squared straight x minus sec squared straight x plus 1 right parenthesis space dx space equals space open square brackets fraction numerator tan cubed straight x over denominator 3 end fraction minus tanx plus straight x close square brackets subscript 0 superscript straight pi over 4 end superscript$
$equals space open parentheses 1 third tan cubed straight pi over 4 minus tan straight pi over 4 plus straight pi over 4 close parentheses space minus space open parentheses 1 third tan cubed 0 minus tan 0 plus 0 close parentheses equals space open parentheses 1 third cross times 1 minus 1 plus straight pi over 4 close parentheses minus left parenthesis 0 plus 0 plus 0 right parenthesis space equals space 1 third minus 1 plus straight pi over 4 equals straight pi over 4 minus 2 over 3$