Integration by Substitution (Reverse Chain Rule)


Integration by Substitution (also called “The Reverse Chain Rule” or “u-Substitution” ) is a method to find an integral, but only when it can be set up in a special way.

It is an important method in mathematics. Integration by substitution is the counterpart to the chain rule for differentiation.

When it is possible to perform an apparently difficult piece of integration by first making a substitution, it has the effect of changing the variable & integrand.

For definite integrals, the limits of integration can also change. For this unit we’ll meet several examples. To master integration by substitution, you need a lot of practice & experience.


Sometimes an apparently sensible substitution doesn’t lead to an integral you will be able to evaluate. So, you need to try out alternative substitutions.


∫ f(g(x)) g′(x) dx = ∫ f(u) du, where u=g(x) and g′(x) dx = du

  • When we can put an integral in this form:

∫ f(g(x)) g′(x) dx

  • Then we can make g(x) = u and g′(x) dx = du
  • Then integrate ∫ f(u) du
  • And finish up by re-inserting g(x) where u is.


f(x) dxuduSubstitutionIntegral of ∫ f(u) duIntegral of ∫ f(x) dx
∫ (x+1)3 dxx+11 dx∫ u3 duIntegration by Substitution Example 1Integration by Substitution Example 2
∫ cos(x2) 2x dxx22x dx∫ cos(u) dusin u + csin x2 + c

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Common Derivatives

Implicit Differentiation

Calculus Index