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fortheloveofscience

  • 2 years ago

integrate the following

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  1. fortheloveofscience
    • 2 years ago
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    \[\int\limits_{0}^{\Pi/2} \sin ^{2}x \div (\sin ^{2}x+4\cos ^{2}x)\]

  2. perl
    • 2 years ago
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    change cos^2 = 1 - sin^2

  3. perl
    • 2 years ago
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    so you have for the denominator sin^2 + 4(1-sin^2) = sin^2 + 4 - 4 sin^2 -3sin^2 + 4

  4. shankvee\
    • 2 years ago
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    I think this is aprtial fractions. If you take cos^2 x common from denominator you have tan^2 x/(tan^2 x+4) Put tan x=t. you'll get dx=dt/sec^2 x= dt/(1+t^2). So you have t^2/((t^2+4)(t^2+1)). Do you know partial fractions?

  5. perl
    • 2 years ago
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    you dont need partional fractions

  6. shankvee\
    • 2 years ago
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    How will you do it by converting denominator to sin?

  7. perl
    • 2 years ago
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    the denominator is -3sin^2 t + 4 , do long division

  8. fortheloveofscience
    • 2 years ago
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    we gottta use the properties of definite integrals

  9. perl
    • 2 years ago
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    shank, change cos^2 = 1 - sin^2

  10. shankvee\
    • 2 years ago
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    And then what?

  11. perl
    • 2 years ago
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    long division

  12. shankvee\
    • 2 years ago
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    ?

  13. perl
    • 2 years ago
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    |dw:1328867815825:dw|

  14. shankvee\
    • 2 years ago
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    You're effectively doing the same thing only.

  15. perl
    • 2 years ago
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    |dw:1328867933423:dw|

  16. shankvee\
    • 2 years ago
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    By doing by partial fractions it reduces to the same thing. Only it is in terms of t^2 so easier to integrate. You directly get tan inverse form which is easy.

  17. perl
    • 2 years ago
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    ok you did trig substitution

  18. shankvee\
    • 2 years ago
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    For the second term you divide by cos ^2 and then you get tan form and then you put that as t to integrate, You get that directly by partial fractions in terms of t.

  19. perl
    • 2 years ago
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    i see that, im asking about your t = tan x part

  20. perl
    • 2 years ago
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    what sort of substitution is this called?

  21. shankvee\
    • 2 years ago
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    Yeah thats a trignometric substituion.

  22. perl
    • 2 years ago
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    ok your solution is better, im not sure if i would go ahead with mine . not sure what do do next

  23. perl
    • 2 years ago
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    you can divide top and bottom by cos^2 for my solution

  24. perl
    • 2 years ago
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    shank, can i ask a simple question about kites

  25. shankvee\
    • 2 years ago
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    hmmm ok.

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