Recall using chain rule in the derivatives unit. You may have used the “u” substitution method where you set a function equal to “u” then substitute that function back in later in the problem. The main objective of this “u” substitution in this integral is to have the “u” function’s derivative’s be another function in this problem.
Say you have the integral ∫ x/(x²+1)dx the “u” function is ALWAYS the one required to be chained (a function within a function), as we can see here, the (x²+1) is the denominator, and can be re-written as (x²+1)⁻¹, making it “a function within a function”.
Substitute this inner function u= (x²+1)
DERIVE “u”, du/dx= 2x. We can see that this 2x is close but not completely exact to the “xdx” . To properly put this du in we need to rearrange the function (via basic algebra), so we divide both sides by 2 to isolate the “x”, then divide both sided by dx. Now we have du/2=xdx.
NOW we can subsituite everything in terms of “u” making ∫ x/(x²+1)dx -> ∫ 1/2u du. Then integrate, plugging the “u=(x²+1)” back in after.
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