Integration by Algebraic SubstitutionExplanation:For an Integral SS=∫f(x) dxmake the algebraic substitutionx=g(u)g′(u)du=dx∴S=∫f(g(u))g′(u)duExampleIntegrate cos(r2)rdrI=∫cos(r2)rdrLet u=r2,du=2rdrI=∫cos(u)2rrdu=∫cos(u)2du=sin(u)2+C=sin(r2)2+C\displaystyle\textsf{Integration by Algebraic Substitution}\\ \textrm{Explanation:}\\ \textsf{For an Integral}\, S\\ S = \int f(x) \, \mathrm{d}x\\ \textsf{make the algebraic substitution}\\x = g(u)\\ g'(u) \mathrm{d}u = \mathrm{d}x\\ \therefore S = \int f(g(u)) g'(u)\mathrm{d}u \\ \textbf{\textsf{Example}}\\ \textsf{Integrate}\, \cos(r^2) r \mathrm{d}r\\ I = \int \cos(r^2) r \mathrm{d}r\\ \textsf{Let}\, u = r^2, \mathrm{d}u = 2r \mathrm{d}r \\ \begin{aligned} I &= \int \frac{\cos(u)}{2r} r \mathrm{d}u \\&= \int \frac{\cos(u)}{2}du \\&= \frac{\sin(u)}{2} + C \\&= \frac{\sin(r^2)}{2} + C \end{aligned}Integration by Algebraic SubstitutionExplanation:For an IntegralSS=∫f(x)dxmake the algebraic substitutionx=g(u)g′(u)du=dx∴S=∫f(g(u))g′(u)duExampleIntegratecos(r2)rdrI=∫cos(r2)rdrLetu=r2,du=2rdrI=∫2rcos(u)rdu=∫2cos(u)du=2sin(u)+C=2sin(r2)+C
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