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6.6 Techniques of Integration (Integration by Substitution) It is often possible to reduce the complexity of an integration problem by making an appropriate change of variable. Let f(x) and g(x) be two given functions and let F(x) be an antiderivative for f(x).The chain rule asserts Turning this formula into an integration formula, we have where C is any constant. If we set f(x) = x3 and g(x) = x2 + 1, then f(g(x)) = (x2 + 1)3 and g’(x) = 2x. By applying aforementioned formula, an antiderivative F(x) of f(x) is given by Now, we have Let us now consider the problem again by using a substitution. Replace g(x) by a new variable u, and replace g’(x) dx with du. We reduce the complexity of f(g(x)) to f(u). Now, Once the antiderivative F(u) is computed, then we make the substitution of g(x) for u and we have Then, The complex integral becomes a simpler one We replace u with g(x) and we are done Integration by Substitution 1.Define a new variable u = g(x), where g(x) is chosen in such a way that, when written in terms of u, the integrand is simpler than when written in terms of x. 2.Transform the integral with respect to x into an integral with respect to u by replacing g(x) everywhere by u and g’(x) dx by du. 3.Integrate the resulting function of u. 4.Rewrite the answer in terms of x by replacing u by g(x).