Say we are solving an indefinite integral ∫x√(2x+1) dx.(adsbygoogle = window.adsbygoogle || []).push({});

According to the textbook, the solution goes like this.

Let u = 2x+1. Then x = (u-1)/2.

Since √(2x+1) dx = (1/2)√u du,

x√(2x+1) dx = [(u-1)/2] * (1/2)√u du.

∫x√(2x+1) dx = ∫[(u-1)/2] * (1/2)√u du. <= What justifies this??

The rest is just trivial calculation.

The only theorem I could rely on here to solve this problem was,

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If u=g(x) is a differentiable function whose range is an interval I, and f is continuous no I, then

∫f(g(x))g'(x) dx = ∫f(u) du.

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Which is so called the substitution rule.

How can this theorem justify the process of the solution I wrote above?

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# Substitution Method on indefinite integral

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