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 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. u = arctan x, v = arcsec x

3 tan u =x, sec v =x. u v Note u = arctan x and v = arcsec x are both defined 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: