How is integration by substitution related to the chain rule?

Question

Emma Aldridge · Accepted Answer

Let #f(x)# be defined and continuous in #[a,b]# and #g(x)# defined and differantiable in #[c,d]# with values in #[a,b]#, such that #g(c) =a# and #g(d) = b#. Suppose also for simplicity that #g'(x) >0#.

The chain rule states that:

#d/dx [f(g(x))] = f'(g(x)) g'(x)#

Consider now the definite integral as limit of the Riemann sum:

# int_c^d f'(g(x)) g'(x)dx = lim_(N->oo) sum_(n=0)^N f(g(xi_k))g'(xi_k)(x_(k+1)-x_k)#

where #x_0 < x_1 < ... < x_N in [c,d]# and #xi_k in (x_k,x_(k+1))#.

As the choice of #xi_k# in the interval is arbitrary we can choose them as the points for which, based on Lagrange's theorem:

#g'(xi_k) = (g(x_(k+1))-g(x_k))/(x_(k+1)-x_k)#

so:

# int_c^d f'(g(x)) g'(x)dx = lim_(N->oo) sum_(n=0)^N f'(g(xi_k))(g(x_(k+1))-g(x_k))#

Let now: #y_k = g(x_k)#. As #g'(x) > 0# the function is strictly increasing so: #y_0 < y_1 <... < y_N#

Then:

# int_a^b f'(g(x)) g'(x)dx = lim_(N->oo) sum_(n=0)^N f'(y_k)(y_(k+1)-y_k)#

but this is a Riemann sum of #f(y)# for #y in [a,b]#, so:

# int_c^d f'(g(x)) g'(x)dx = int_a^b f'(y) dy = f(b)-f(a)#

Emilia Aguilar · Answer

undefined Integration by substitution and the chain rule in calculus are related concepts. Integration by substitution is essentially the inverse operation of the chain rule.

In integration by substitution, we make a substitution of variables to rewrite the integral in terms of a new variable, making the integration process easier. This substitution often involves composing the function inside the integral with another function, similar to how the chain rule works.

The chain rule states how to differentiate composite functions, where one function is applied to the output of another function. It enables us to find the derivative of the composite function by taking the derivative of the outer function and then multiplying it by the derivative of the inner function.

Integration by substitution reverses this process. It involves breaking down a complicated integral into simpler parts by replacing a variable or expression with another one, essentially "unwinding" the composition introduced by the chain rule. This substitution simplifies the integral and allows us to find the antiderivative more easily.