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Simplifying an antiderivative

  1. Jan 12, 2016 #1
    Mod note: Moved from a homework section
    1. The problem statement, all variables and given/known data

    Hello my question more has to do with theory that perhaps deals with algebra.

    Why is the following true?

    7b9324fbb148a9401897d88800b7d547.png

    2. Relevant equations
    N/A

    3. The attempt at a solution
    N/A
     
    Last edited by a moderator: Jan 12, 2016
  2. jcsd
  3. Jan 12, 2016 #2

    fresh_42

    Staff: Mentor

    Have you tried to apply the formulas ##cos^2x + sin^2x = 1## and ##tan (x)=\frac{sin (x)}{cos (x)}##?
     
  4. Jan 12, 2016 #3
    Hello, fresh_42

    Thank you for the reply I have attempted to apply the identity cos^2x + sin^2x = 1 and could not figure out an algebraic means to have this work out. As for the reciprocal identity tanx = sinx/cosx. I do not see how this identity applies to arctanx since arctanx does not equal arcsinx/arccosx

    Regards,

    OmniNewton
     
  5. Jan 12, 2016 #4

    Mark44

    Staff: Mentor

    Because you have integrals on both sides, you can instead show that the two integrands are equal (plus possibly a constant). IOW, just show that ##\sin^2(\arctan(x)) = \frac{x^2}{x^2 + 1} + C##

    The best way to do this, IMO, is to draw a right triangle, and label an acute angle ##\theta##, with the opposite as x and the adjacent side as 1. The ##\tan(\theta) = \frac x 1##. Now find ##sin(\theta)##.
     
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