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61.
Evaluate the following definite integral:
$integral subscript 0 superscript 2 fraction numerator 5 straight x plus 1 over denominator straight x squared plus 4 end fraction dx$

Let I = $integral subscript 0 superscript 2 fraction numerator 5 straight x plus 1 over denominator straight x squared plus 4 end fraction dx space equals space integral subscript 0 superscript 2 fraction numerator 5 straight x over denominator straight x squared plus 4 end fraction dx plus integral subscript 0 superscript 2 fraction numerator 1 over denominator straight x squared plus 4 end fraction dx$

$equals space 5 over 2 integral subscript 0 superscript 2 fraction numerator 2 straight x over denominator straight x squared plus 4 end fraction dx plus integral subscript 0 superscript 2 fraction numerator 1 over denominator straight x squared plus left parenthesis 2 right parenthesis squared end fraction dx$

$equals space 5 over 2 open square brackets log left parenthesis straight x squared plus 4 right parenthesis close square brackets subscript 0 superscript 2 space plus space 1 half open square brackets tan to the power of negative 1 end exponent straight x over 2 close square brackets subscript 0 superscript 2$

$open square brackets because space integral fraction numerator straight f apostrophe left parenthesis straight x right parenthesis over denominator straight f left parenthesis straight x right parenthesis end fraction dx space equals space log open vertical bar straight f left parenthesis straight x right parenthesis close vertical bar space and space integral fraction numerator 1 over denominator straight x squared plus straight a squared end fraction dx space equals space 1 over straight a tan to the power of negative 1 end exponent straight x over straight a close square brackets$

$equals space 5 over 2 left square bracket log space 8 minus space log space 4 right square bracket plus 1 half left square bracket tan to the power of negative 1 end exponent 1 space minus space tan to the power of negative 1 end exponent 0 right square bracket space equals space 5 over 2 log open parentheses 8 over 4 close parentheses plus 1 half open square brackets straight pi over 4 minus 0 close square brackets equals space 5 over 2 log space 2 space plus space straight pi over 8$

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