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Example 10 Find

Example 10 Find Solution Begin by using the additive property to write the given integral as the sum of two integrals, each with one constant bound: Before differentiating, interchange the bounds of the first integral:

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Example 10 Find

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  1. Example 10 Find Solution Begin by using the additive property to write the given integral as the sum of two integrals, each with one constant bound: Before differentiating, interchange the bounds of the first integral: Let u = arctan x, v = arcsec x and By the chain rule:

  2. u = arctan x, v = arcsec x By the First Fundamental Theorem of Calculus, the derivative F /(w) is obtained by replacing t by w in the integrand sin t: Apply this formula for w=u and for w=v: Use the triangles on the next slide to simplify this expression.

  3. tan u=x, sec v=x. v u Note u = arctan x and v = arcsec x are bothdefined when x  (-,-1]  [1, ). Hence u  (- /2,- /4 ]  [ /4,  /2) while v  [0, /2 )  ( /2,  ]. Thus sin u and tan u = x have the same sign while sin v is positive. From the triangles above:

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