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Math Help - Techniques of integration (2)

  1. #1
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    Techniques of integration (2)

    Evaluate \int \frac{x^2}{4x \sin x + (4 - x^2) \cos x} \ dx.


    Hint:

    Spoiler:

    divide top and botttom of the integrand by "something" and then make a clever trig substitution!
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  2. #2
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    I just give the final answer and the important step :

    The final answer is - arccosh| cosec { x + arctan [(4-x^2 )/ ( 4x )] } | + c

    And the important step is

    4x sinx + ( 4 - x^2 ) cosx =
    sqrt [ (4x)^2 + (4-x^2)^2 ] * sin { x + arctan[ (4-x^2)/(4x) ] }
    = sqrt [ x^4 + 8x^2 + 16 ] * sin { x + arctan[ (4-x^2)/(4x) ] }
    = ( x^2 + 4 ) * sin { x + arctan[ (4-x^2)/(4x) ] }
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  3. #3
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    Quote Originally Posted by simplependulum View Post

    4x \sin x + ( 4 - x^2 ) \cos x = \sqrt{ (4x)^2 + (4-x^2)^2 } \ \sin \{ x + \arctan [(4-x^2)/(4x) ] \}

    = \sqrt{x^4 + 8x^2 + 16} \ \sin \{ x + \arctan[ (4-x^2)/(4x) ] \}

    = ( x^2 + 4 ) \sin \{ x + \arctan[ (4-x^2)/(4x) ] \}
    correct! then the substitution x + \arctan[ (4-x^2)/(4x) ]=t will reduce the integral to the simple integral \int \csc t \ dt.
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