(vi.)

∫ dx= log (tan x + sec x).
cos x

(vii.) ∫ eax sin (bx + α) dx = (a2 + b2)−1 eax {a sin (bx + α) − b cos (bx + α) }.

(viii.) ∫ sinm x cosn x dx can be reduced by differentiating a function of the form sinp x cosq x;

e.g. d sin x= 1+ q sin2 x= 1 − q+ q.
dx cosq xcosq−1 x cosq+1 xcosq−1 x cosq+1 x

Hence

∫ dx= sin x+ n − 2∫ dx.
cosn x (n − 1) cosn−1 xn − 1 cosn−2 x

(ix.)

∫1/2π0 sin2n x dx = ∫1/2π0 cos2n x dx = 1·3 ... (2n − 1)· π, (n an integer).
2·4 ... 2n 2

(x.)