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By far the most important tool of anti-differentiation is that of applying the chain rule of differentiation backwards; that means given a function f(x), finding a function u(x), so that you can write f(x) as In that case you can claim that an anti-derivative of f is g(u(x)).

For example, suppose we have f(x) = sin⁵ x cos x.
Then we can set u(x )= sin x, and find that , which we can recognize as the derivative of , which is then an anti-derivative of f. The identical idea can be used to anti-differentiate any polynomial in sines and cosines of odd degree in all, if you apply the identity (sin²x + cos²x = 1) to bring all terms into forms that are linear in one of sine or cosine like f is above.

Employing this technique is called "substituting"; the variable u is in a sense substituted for x. Notice that if we can recognize that f(x) is the derivative of a known function then we can anti-differentiate by inspection. By attempting to substitute u(x) we change the question to: can we recognize as the derivative of a function of u?
In the example above, the new question becomes: can we recognize sin⁵ x as a derivative with respect to the variable sin x? Since sin⁵ x is a power of sin x, we can recognize it as the derivative of and we have found an anti-derivative of f.
There are some well known standard substitutions that allow us to anti-differentiate many potentially useful functions. They are listed in the chart below.

Looking for the right substitution that might help for finding some given specific anti-derivative is a bit of intellectual detective work. It is akin to solving a chess problem, in that while perhaps it is of no practical value, it may well be an excellent exercise for developing your reasoning powers or your ability to find what at first seems like finding a needle in a haystack.