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