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

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space sin space 3 straight x space dx space equals space 1 half integral subscript 0 superscript straight pi over 4 end superscript left parenthesis space 2 space sin space 3 straight x space sin space 2 straight x right parenthesis space dx$
$equals space 1 half integral subscript 0 superscript straight pi over 4 end superscript left square bracket space cos space left parenthesis 3 straight x minus 2 straight x right parenthesis space minus space cos space left parenthesis 3 straight x plus 2 straight x right parenthesis right square bracket space dx space equals space 1 half integral subscript 0 superscript straight pi over 4 end superscript left parenthesis cosx space minus space cos space 5 straight x right parenthesis space dx$
$equals space 1 half open square brackets sin space straight x minus space fraction numerator sin space 5 straight x over denominator 5 end fraction close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space 1 half open square brackets open parentheses sin space straight pi over 4 minus 1 fifth sin fraction numerator 5 straight pi over denominator 4 end fraction close parentheses minus open parentheses sin space 0 minus 1 fifth sin space 0 close parentheses close square brackets$
$equals 1 half open square brackets open parentheses sin space straight pi over 4 plus 1 fifth sin straight pi over 4 close parentheses minus open parentheses sin space 0 space minus space 1 fifth sin space 0 close parentheses close square brackets equals space 1 half open square brackets open parentheses fraction numerator 1 over denominator square root of 2 end fraction plus 1 fifth cross times fraction numerator 1 over denominator square root of 2 end fraction close parentheses minus left parenthesis 0 minus 0 right parenthesis close square brackets space equals space 1 half open square brackets 6 over 5 cross times fraction numerator 1 over denominator square root of 2 end fraction close square brackets space equals space fraction numerator 3 over denominator 5 square root of 2 end fraction$

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