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Question Number 141496 by cesarL last updated on 19/May/21

$$\int \frac{\sqrt{2+9x^2}}{x}dx$$$$SOSSOSHELP$$

Answered by qaz last updated on 19/May/21

$$\int \sqrt{2+9x^2}\frac{dx}{x}$$$$=\frac{1}{2}\int \sqrt{2+9x^2}\frac{d\left(x^2\right)}{x^2}$$$$=\frac{1}{2}\int \sqrt{2+9y}\frac{dy}{y}................x^2=y$$$$=\frac{1}{2}\int \sqrt{2(1+\frac{9}{2}y)}\frac{dy}{y}$$$$=\sqrt{2}\int \sec {}^{3}t\cdot \frac{dt}{\tan t}..............\frac{9}{2}y={\tan }^{2}t.....dy=\frac{4}{9}\tan t\sec {}^{2}tdt$$$$=\sqrt{2}\int \frac{dt}{\cos {}^{2}t\sin t}..............\cos t=\frac{\sqrt{2}}{\sqrt{9y+2}}$$$$=\sqrt{2}\int \frac{d\left(\cos t\right)}{(\cos {}^{2}t-1)\cos {}^{2}t}$$$$=\sqrt{2}\int (\frac{1}{\cos {}^{2}t-1}-\frac{1}{\cos {}^{2}t})d\left(\cos t\right)$$$$=\frac{1}{\sqrt{2}}ln\mid \frac{\cos t-1}{\cos t+1}\mid +\frac{\sqrt{2}}{\cos t}+C$$$$=\frac{1}{\sqrt{2}}ln\mid \frac{\frac{\sqrt{2}}{\sqrt{9x^2+2}}-1}{\frac{\sqrt{2}}{\sqrt{9x^2+2}}+1}\mid +\sqrt{9x^2+2}+C$$

Answered by MJS_new last updated on 19/May/21

$$\int \frac{\sqrt{a^2{x}^2+b^2}}{x}dx=$$$$[x=\frac{b(t^2-1)}{2at}\iff t=\frac{ax+\sqrt{a^2{x}^2+b^2}}{b}\to dx=\frac{b\sqrt{a^2{x}^2+b^2}}{a(ax+\sqrt{a^2{x}^2+b^2})}dt]$$$$=\frac{b}{2}\int \frac{{(t^2+1)}^2}{t^2(t^2-1)}dt=$$$$=\frac{b}{2}\int \frac{t^2-1}{t^2}dt+b\int \frac{dt}{t-1}-b\int \frac{dt}{t+1}=$$$$=\frac{b(t^2+1)}{2t}+b\ln \frac{t-1}{t+1}=$$$$=\sqrt{a^2{x}^2+b^2}+b\ln \mid \frac{b-\sqrt{a^2{x}^2+b^2}}{x}\mid +C$$

Answered by mathmax by abdo last updated on 20/May/21

$$\mathrm{\Psi }=\int \frac{\sqrt{2+{9\mathrm{x}}^2}}{\mathrm{x}}\mathrm{dx}\Rightarrow \mathrm{\Psi }{=}_{3\mathrm{x}=\sqrt{2}\mathrm{sht}}3\int \frac{\sqrt{2}\mathrm{ch}\left(\mathrm{t}\right)}{\sqrt{2}\mathrm{sht}}\times \frac{\sqrt{2}}{3}\mathrm{cht}\mathrm{dt}$$$$=\sqrt{2}\int \frac{{\mathrm{ch}}^2\mathrm{t}}{\mathrm{sht}}\mathrm{dt}=\frac{\sqrt{2}}{2}\int \frac{1+\mathrm{ch}\left(2\mathrm{t}\right)}{\mathrm{sht}}\mathrm{dt}$$$$=\frac{\sqrt{2}}{2}\int \frac{1+\frac{{\mathrm{e}}^{2\mathrm{t}}+{\mathrm{e}}^{-2\mathrm{t}}}{2}}{\frac{{\mathrm{e}}^{\mathrm{t}}-{\mathrm{e}}^{-\mathrm{t}}}{2}}\mathrm{dt}=\frac{\sqrt{2}}{2}\int \frac{2+{\mathrm{e}}^{2\mathrm{t}}+{\mathrm{e}}^{-2\mathrm{t}}}{{\mathrm{e}}^{\mathrm{t}}-{\mathrm{e}}^{-\mathrm{t}}}\mathrm{dt}$$$${=}_{{\mathrm{e}}^{\mathrm{t}}=\mathrm{y}}\frac{1}{\sqrt{2}}\int \frac{2+{\mathrm{y}}^2+{\mathrm{y}}^{-2}}{\mathrm{y}-{\mathrm{y}}^{-1}}\frac{\mathrm{dy}}{\mathrm{y}}\Rightarrow \sqrt{2}\mathrm{\Psi }=\int \frac{{\mathrm{y}}^2+{\mathrm{y}}^{-2}+2}{{\mathrm{y}}^2-1}\mathrm{dy}$$$$=\int \frac{{\mathrm{y}}^4+1+{2\mathrm{y}}^2}{{\mathrm{y}}^2({\mathrm{y}}^2-1)}\mathrm{dy}=\int \frac{{\mathrm{y}}^4+{2\mathrm{y}}^2+1}{{\mathrm{y}}^4-{\mathrm{y}}^2}\mathrm{dy}$$$$=\int \frac{{\mathrm{y}}^4-{\mathrm{y}}^2+{3\mathrm{y}}^2+1}{{\mathrm{y}}^4-{\mathrm{y}}^2}\mathrm{dy}=\mathrm{y}+\int \frac{{3\mathrm{y}}^2+1}{{\mathrm{y}}^2({\mathrm{y}}^2-1)}\mathrm{dy}$$$$\mathrm{let}\mathrm{decompose}\mathrm{F}\left(\mathrm{y}\right)=\frac{{3\mathrm{y}}^2+1}{{\mathrm{y}}^2({\mathrm{y}}^2-1)}=\frac{{3\mathrm{y}}^2+1}{{\mathrm{y}}^2(\mathrm{y}-1)(\mathrm{y}+1)}$$$$\mathrm{F}\left(\mathrm{y}\right)=\frac{\mathrm{a}}{\mathrm{y}}+\frac{\mathrm{b}}{{\mathrm{y}}^2}+\frac{\mathrm{c}}{\mathrm{y}-1}+\frac{\mathrm{d}}{\mathrm{y}+1}$$$$\mathrm{b}=-1,\mathrm{c}=\frac{4}{2}=2,\mathrm{d}=\frac{4}{-2}=-2\Rightarrow \mathrm{F}\left(\mathrm{y}\right)=\frac{\mathrm{a}}{\mathrm{y}}-\frac{1}{{\mathrm{y}}^2}+\frac{2}{\mathrm{y}-1}-\frac{2}{\mathrm{y}+1}$$$$\mathrm{F}\left(2\right)=\frac{13}{12}=\frac{\mathrm{a}}{2}-\frac{1}{4}+2-\frac{2}{3}=\frac{\mathrm{a}}{2}-\frac{1}{4}+\frac{4}{3}=\frac{\mathrm{a}}{2}+\frac{13}{12}\Rightarrow \mathrm{a}=0\Rightarrow$$$$\int \mathrm{F}\left(\mathrm{y}\right)\mathrm{dy}=\frac{1}{\mathrm{y}}+2\log \mid \frac{\mathrm{y}-1}{\mathrm{y}+1}\mid +\mathrm{C}$$$$={\mathrm{e}}^{-\mathrm{t}}+2\log \mid \frac{{\mathrm{e}}^{\mathrm{t}}-1}{{\mathrm{e}}^{\mathrm{t}}+1}\mid +\mathrm{C}\mathrm{but}\mathrm{t}=\mathrm{argsh}\left(\frac{3\mathrm{x}}{\sqrt{2}}\right)$$$$=\log (\frac{3\mathrm{x}}{\sqrt{2}}+\sqrt{1+\frac{{9\mathrm{x}}^2}{2}})\Rightarrow$$$$\sqrt{2}\mathrm{\Psi }=\frac{3\mathrm{x}}{\sqrt{2}}+\sqrt{1+\frac{{9\mathrm{x}}^2}{2}}+2\log \mid \frac{\frac{3\mathrm{x}}{\sqrt{2}}+\sqrt{1+\frac{{9\mathrm{x}}^2}{2}}-1}{\frac{3\mathrm{x}}{\sqrt{2}}+\sqrt{1+\frac{{9\mathrm{x}}^2}{2}}+1}\mid +\mathrm{C}\Rightarrow$$$$\mathrm{\Psi }=\frac{1}{2}(3\mathrm{x}+\sqrt{2+{9\mathrm{x}}^2})+\sqrt{2}\log \mid \frac{3\mathrm{x}+\sqrt{2+{9\mathrm{x}}^2}-\sqrt{2}}{3\mathrm{x}+\sqrt{2+{9\mathrm{x}}^2}+\sqrt{2}}\mid +\mathrm{C}$$

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