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

I = $integral subscript 1 superscript 2 left parenthesis logx squared right parenthesis space dx space space space space equals 2 space integral subscript 1 superscript 2 left parenthesis log space straight x right parenthesis. space space space 1 space dx space equals space 2 space open curly brackets open square brackets left parenthesis logx right parenthesis. straight x close square brackets subscript 1 superscript 2 minus integral subscript 1 superscript 2 1 over straight x. straight x space dx close curly brackets$
$equals space 2 left parenthesis 2 space log space 2 space minus space 1 space log space 1 right parenthesis space minus 2 integral subscript 1 superscript 2 1 space dx space equals space 2 left parenthesis 2 space log space 2 space minus 0 right parenthesis space minus space 2 open square brackets straight x close square brackets subscript 1 superscript 2 equals space 4 space log space 2 space minus 2 space left parenthesis 2 minus 1 right parenthesis space equals space 4 space log space 2 space minus space 2$

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