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In this tutorial, we will discuss the use of substitution when doing integration.
Substitution
Substitution can be viewed as the reverse of the Chain Rule for derivatives.
) g\prime(x) \, dx=\int f(u) \, du "\int f(g(x)) g\prime(x) \, dx=\int f(u) \, du")
where  "u=g(x)")
Examples
Find the anti-derivative of the following functions using substitution
^2\,dx "1. \,\, \int 2x(x^2+1)^2\,dx")
now we need to make our choice for 'u', as a general rule 'u' is usually the more complicated function (although this is not a good rule of thumb to always go by). The true reason we choose
is because
This will simplify the integral to
pluggin 'u' back in and we get
^3}{3}+C "=\frac{(x^2+1)^3}{3}+C")
) sin(x)\, dx "2. \,\, \int cos(cos(x)) sin(x)\, dx")
Now let us choose u, in this case there is not really 'a more complicated function' as in the last example. We have to be a little clever, and choose the function such that when we take 'u' and the derivative of 'u' it will simplify the integral. An appropriate choice would be
 "u = cos(x)")
because
 \, \to \, -du = sin(x) "du = -sin(x) \, \to \, -du = sin(x)")
this will simplify the integral to
\, du = -sin(u)+C = -sin(cos(x))+C "\int -cos(u)\, du = -sin(u)+C = -sin(cos(x))+C")
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