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[Note:] The error described here has been corrected in the meantime in response to this question.

When checking wikipedia on substition they say that \int f(g(t))g'(t) dt = \int f(x)dx with x = g(t). Which is true. However, when checking example 1 on the page ( http://en.wikipedia.org/wiki/Integration_by_substitution)

They say that when evaluating $\int x\cos(x^2 +1) dx$ in example one, they use the substition $u=x^2 + 1$ and say that they use the rule mentioned above from right to left, however this isn't the case right? Aren't they using that rule from left to right?

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    @Bill: No, they really did mean the direction of the equation, which was the other way around five years ago -- see my answer.2012-01-22

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You're quite right – well spotted; this error has been there for almost five years! It was introduced by this edit, which switched the direction of the equation but failed to adapt the examples. I've now adapted the examples.