Let u= g(x) , where g\' is continuous on an interval, and let f be continuous on

Question

Let u= g(x) , where g' is continuous on an interval, and let f be continuous on The corresponding range of g . On that interval, f (g(x))g'(x)dx = f(u)d. (Substitution Rule for Indefinite s) Let u = cos.x . Then, one has du = (cos.x) ax du = - sin xdx en dash du = sin xdx Substitution Rule for Indefinite s shows th sin3 xdx = sin2 .v(sin xdx) = (l - cos2 x) (sin xdx) = (1-u2)(-du) = (u2-1)du Continue The computation. sin'xdx = (u2 -1)du = 1/3 u3-u+C =1/3cos3 x-cosx+C Hence, sin3 xdx=1/3cos3 x-cos x+C

Explanation / Answer

1 is not taken into account because the substituttion is cosx=u

had the substitution been u = 1-cosx we would have taken 1 into account then

its just the matter of what function is subsatituted

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