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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript space straight x squared space sinx space dx space equals space open square brackets straight x squared left parenthesis negative left parenthesis cosx right parenthesis close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space integral subscript 0 superscript straight pi divided by 2 end superscript space 2 straight x space left parenthesis negative cos space straight x right parenthesis space dx$
$equals space minus space open square brackets straight x squared space cosx close square brackets subscript 0 superscript straight pi divided by 2 end superscript space plus space 2 space integral subscript 0 superscript straight pi divided by 2 end superscript straight x space cosx space dx$
$equals negative open square brackets straight x squared space cosx close square brackets subscript 0 superscript straight pi divided by 2 end superscript plus 2 open curly brackets open square brackets straight x space sinx close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space integral subscript 0 superscript straight pi over 2 end superscript space 1. space sin space straight x space dx close curly brackets$
$equals negative open square brackets straight x squared space cosx close square brackets subscript 0 superscript straight pi divided by 2 end superscript space plus space 2 open square brackets straight x space sinx close square brackets subscript 0 superscript straight pi divided by 2 end superscript space plus space 2 open square brackets cos space straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript equals space minus open square brackets straight pi squared over 4 cos straight pi over 2 minus 0 close square brackets space plus space 2 open square brackets straight pi over 2 space sin straight pi over 2 minus 0 close square brackets space plus space 2 open square brackets cos space straight pi over 2 minus cos space 0 close square brackets equals space minus left parenthesis 0 minus 0 right parenthesis plus 2 open parentheses straight pi over 2 minus 0 close parentheses plus 2 space left parenthesis 0 minus 1 right parenthesis space equals space 0 plus straight pi minus 2 space equals space straight pi minus 2.$

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