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

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66.
Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$
Let $straight I subscript 1 space equals space integral space sin cubed space 2 straight t space cos space 2 straight t space dt$
Put sin 2t = y; $therefore space space 2 space cos space 2 dt space dt space equals space dy space space rightwards double arrow space space space space cos space 2 straight t space dt space equals space 1 half dy$
$therefore$ $straight I subscript 1 space equals space 1 half integral straight y cubed dy space equals space 1 half cross times straight y to the power of 4 over 4 space equals space 1 over 8 sin to the power of 4 2 straight t space equals space straight F left parenthesis straight t right parenthesis comma space say.$
By second fundamental theorem,
$straight I space equals space straight F open parentheses straight pi over 4 close parentheses minus straight F left parenthesis 0 right parenthesis space equals space 1 over 8 sin to the power of 4 straight pi over 2 minus 1 over 8 sin to the power of 4 0 space equals space 1 over 8 left parenthesis 1 right parenthesis to the power of 4 minus 1 over 8 left parenthesis 0 right parenthesis to the power of 4 space equals space 1 over 8 minus 0 space equals space 1 over 8$

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