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Integrals

Short Answer Type

51.

Explain the following integral:
$space integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight x squared end fraction dx$

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52.

Explain the following integral:
$integral subscript 2 superscript 3 fraction numerator 1 over denominator straight x squared minus 1 end fraction dx$

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53.
Explain the following integral:
$integral subscript 2 superscript 3 fraction numerator straight x over denominator straight x squared plus 1 end fraction dx$

Let I = $integral subscript 2 superscript 3 fraction numerator straight x over denominator straight x squared plus 1 end fraction dx space equals space 1 half integral subscript 2 superscript 3 fraction numerator 2 straight x over denominator straight x squared plus 1 end fraction dx space equals space 1 half left square bracket log left parenthesis straight x squared plus 1 right parenthesis right square bracket subscript 2 superscript 3 space space space$
$equals space 1 half left parenthesis log space 10 space minus space log space 5 right parenthesis space equals space 1 half log open parentheses 10 over 5 close parentheses space equals space 1 half log 2$

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54. Evaluate the following integral:

integral subscript 1 superscript 2 fraction numerator dx over denominator left parenthesis straight x plus 1 right parenthesis space left parenthesis straight x plus 2 right parenthesis end fraction

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55. Evaluate the following integral:

integral subscript 1 superscript 2 fraction numerator 5 straight x squared over denominator straight x squared plus 4 straight x plus 3 end fraction dx

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56. Evaluate the following integral

space integral subscript 2 superscript 3 fraction numerator 1 over denominator straight x squared minus 1 end fraction dx

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57. Evaluate the following integral:

space space integral subscript 0 superscript 4 fraction numerator dx over denominator square root of straight x squared plus 2 straight x plus 3 end root end fraction

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58. Evaluate the following integral:

integral subscript 4 superscript 9 fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx

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59.

Evaluate the following definite integral
$integral subscript 0 superscript 2 fraction numerator 6 straight x plus 3 over denominator straight x squared plus 4 end fraction dx.$

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60.

Evaluate the following definite integral
$integral subscript 0 superscript 1 fraction numerator 2 straight x over denominator 5 straight x squared plus 1 end fraction dx$