Some Important Integral Formulas

integral formulas shahbaz ahmed June 2024 Some important integral formulas sinn ax dx R In = sinn−1 ax(sin ax dx) In = R Now d(cos ax) = − sin axd(ax) = −a sin ax dx d(cos ax) = −a sin ax dx − a1 d(cos ax) = sin ax dx Putting in the above integral R In − sinn−1 ax[− a1 d(cos ax)] R In = − a1 sinn−1 ax[d(cos ax)] R In = − a1 [sinn−1 ax(cos ax) − cos ax[d(sinn−1 ax)] d(sinn−1 ax) = (n − 1) sinn−2 ax d[sin ax] d(sinn−1 ax) = (n − 1) sinn−2 ax cos ax d(ax) d(sinn−1 ax) = a(n − 1) sinn−2 ax cos ax dx d(sinn−1 ax) = a(n − 1) cos ax sinn−2 ax dx Putting in the above integral In = − a1 [sinn−1 ax (cos ax) − cos ax[a(n − 1) cos ax sinn−2 ax dx]] R In = − a1 [sinn−1 ax (cos ax) − a(n − 1) (cos2 ax sinn−2 ax dx)] R In = − a1 [sinn−1 ax (cos ax) − a(n − 1) (1 − sin2 ax) sinn−2 ax dx] R In = − a1 sinn−1 ax (cos ax) + (n − 1) (1 − sin2 ax) sinn−2 ax dx R 1 R R In = − a1 sinn−1 ax (cos ax)+(n−1) sinn−2 ax dx− (n−1) sinn ax R In = − a1 sinn−1 ax (cos ax) + (n − 1) sinn−2 ax dx − (n − 1)In R In + (n − 1)In = − a1 sinn−1 ax (cos ax) + (n − 1) sinn−2 ax dx− R [1 + (n − 1)]In = − a1 sinn−1 ax (cos ax) + (n − 1) sinn−2 ax dx− R nIn = − a1 sinn−1 ax (cos ax) + (n − 1) sinn−2 ax dx− R n−1 sinn axdx = − sin ax cos ax na + n−1 n R sinn−2 ax dx ....................................................................... In = R cosn ax dx In = R cosn−1 ax cos ax dx Now d(sin ax) = cos ax d(ax) = a cos ax dx 1 a d(sin ax) = cos ax dx Putting in the above integral R In = a1 cosn−1 ax d(sin ax) R In = a1 [cosn−1 ax sin ax − sin ax d(cosn−1 ax)] Now d(cosn−1 ax) = (n − 1) cosn−2 ax d(cos ax) d(cosn−1 ax) = −(n − 1) cosn−2 ax (sin ax) d(ax) d(cosn−1 ax) = −a(n − 1) cosn−2 ax (sin ax) dx Putting in the above integral R In = a1 [cosn−1 ax sin ax − sin ax [−a(n − 1) cosn−2 ax (sin ax) dx]] R In = a1 [cosn−1 ax sin ax + a(n − 1) (sin2 ax) cosn−2 ax dx] R In = a1 [cosn−1 ax sin ax + a(n − 1) (1 − cos2 ax) cosn−2 ax dx] 2 dx In = 1 a cosn−1 ax sin ax + (n − 1) R 1 a cosn−1 ax sin ax + (n − 1) R cosn−2 ax R dx − (n − 1) cosn ax dx cosn−2 ax dx − (n − 1)In R In + (n − 1)In = a1 cosn−1 ax sin ax + (n − 1) cosn−2 ax dx R [1 + n − 1]In = a1 cosn−1 ax sin ax + (n − 1) cosn−2 ax dx R cosn−2 ax dx nIn = a1 cosn−1 ax sin ax + (n − 1) R 1 In = an cosn−1 ax sin ax + n−1 cosn−2 ax dx n R R 1 cosn ax dx = an cosn−1 ax sin ax + n−1 cosn−2 ax dx n In = 3