A Slick Substitution
If the function “cos sqrt(x) dx” has to be integrated, would you consider using the substitution method?
Let w = sqrt(x), dw = dx/2sqrt(x), or 2dw = dx/sqrt(x) . . .
Are we stuck? Typically, we employ the substitution method for integration when we can identify a function and its derivative within the integrand. For example, in integral[cos x sin x dx], we recognize the function “sin x” and its derivative “cos x dx”, so the substitution methodology here is straightforward. We let u = sin x and du = cos x dx.
However, for integral[cos sqrt(x) dx] we must be creative and remember that w = sqrt(x), then 2dw = dx/sqrt(x) = dx/w, or 2wdw = dx.* Now, integral[cos sqrt(x) dx] = integral[2w cos w dw] = 2 integral[w cos w dw.] This integral is now fit for integration-by-parts using the “Liate” rule where we let “u” equal the algebraic vs the trigonometric function. Then we conclude by reversing the substitution. Let u = w, dv = cos w dw, so du = dw and v = sin w.
integral[cos sqrt(x) dx] = 2 integral[w cos w dw] = 2(w sin w - integral[sin w dw]) = 2(w sin w + cos w) + c = 2[sqrt(x) sin sqrt(x) + cos sqrt(x)] + c.
We check our result by differentiation: d/dx[2sqrt(x) sin sqrt(x) + 2cos sqrt(x) + c] = [1/sqrt(x)] sin sqrt(x) + [1/sqrt(x)] sqrt(x) cos sqrt(x) – [1/sqrt(x)] sin sqrt(x) = cos sqrt(x).
*Thanks “DavidG” from “YAHOO! Answers” who completed an undergraduate degree in mathematics in 2005, worked for a few years and may have already finished a Masters of Information Technology (focusing on Computer Science) from The University of Northern Iowa.