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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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    I think they just meant that they replaced the right hand side "$x^2+1$" with the left hand side "$u$". It's not worded particularly well.2012-01-22
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    Good catch there! I've noticed some integration by substitution mistakes in some main calculus textbooks such as Larson in which they don't actually give the notation for "let $x=g(t)$" and they simply say that an expression in terms of $x$ and an expression in terms of $t$ are equal.2012-01-22
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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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