452 Trigonometric functions
Students are frequently called upon to realize that the formula
/1 +cos2x 2 1 + 2 cos2x +cost2x
cos4 x = (cost x)2 = ( 2 = 4is useful when the left member must be integrated, and to know what to
do to complete the problem.
The integral in(8.26) I = f sin nix cos nx dxlooks forbidding until it is discovered or remembered that a useful
formula for the integrand can be obtained by adding the formulassin (8+0)=sin8cos4'+cos0sin4'
sin (8-0) =sin8cos¢-cos8sinc'.
Thus
11 [sin (mx + nx) + sin (mx - nx)J = sin mx cos nx
and(8.261) I =
2(m+n)f [sin (m + n)xJ(m + n) dx
+^1 [sin (m - n)x] (m - n) dx
2(m-n)= 2(m + n)
cos (m + n)x -
2(m1
n)cos (m - n)x + c
except when m = n or m = - n. Similarly, the integrals
(8.262) f sin mx sin nx dx, f cos mx cos nx dxcan be evaluated by use of formulas obtained by adding and subtracting
the formulas
cos
(8 + cos cos 95 - sin 8 sin 0.Sometimes integrals can be evaluated by making quite direct use of
the power formula and other integration formulas. For example,(8.'63) f siriE. uxc-s-sx..4
when a ,E 0 and p s -1. Sometimes we need some ingenuity. For
example, the integral(8.264) f sin' x cos' x dx