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Question Number 72494 by aliesam last updated on 29/Oct/19

$$\frac{\int x\sqrt{x^2+5}dx-3\int \frac{x}{\sqrt{x^2+5}}dx}{\int \frac{x(x^2+2)}{\sqrt{x^2+5}}dx}$$

Commented by mathmax by abdo last updated on 29/Oct/19

$$letI=\int x\sqrt{x^2+5}dx,J=\int \frac{xdx}{\sqrt{x^2+5}}$$$$K=\int \frac{x(x^2+2)}{\sqrt{x^2+5}}dxwehaveI-J=\int \frac{x(x^2+5)-3x}{\sqrt{x^2+5}}dx$$$$=\int \frac{x^3+5x-3x}{\sqrt{x^2+5}}dx=\int \frac{x(x^2+2)}{\sqrt{x^2+5}}dx\Rightarrow \frac{I-3J}{K}=1$$

Answered by Tanmay chaudhury last updated on 29/Oct/19

$$I=\frac{I_1-3I_2}{I_3}$$$$I_1=\int x\sqrt{x^2+5}dx$$$$=\frac{1}{2}\int {(x^2+5)}^{\frac{1}{2}}\times d(x^2+5)$$$$=\frac{1}{2}\times \frac{{(x^2+5)}^{\frac{3}{2}}}{\frac{3}{2}}+c_1$$$$I_2=\int \frac{x}{\sqrt{x^2+5}}dx$$$$=\frac{1}{2}\int {(x^2+5)}^{\frac{-1}{2}}\times d(x^2+5)$$$$=\frac{1}{2}\times \frac{{(x^2+5)}^{\frac{1}{2}}}{\frac{1}{2}}+c_2$$$$\int \frac{x(x^2+2)}{\sqrt{x^2+5}}dx$$$$k^2=x^2+5\to 2kdk=2xdx$$$$\int \frac{k^2-5+2}{k}\times kdk$$$$\int k^2-3dk$$$$\frac{k^3}{3}-3k+c_3$$$$\frac{{(x^2+5)}^{\frac{3}{2}}}{3}-3{(x^2+5)}^{\frac{1}{2}}+c_3$$$$\mathit{n}\mathit{o}\mathit{w}\mathit{p}\mathit{l}\mathit{s}\mathit{p}\mathit{u}\mathit{t}\mathit{t}\mathit{h}\mathit{e}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{o}\mathit{g}\mathit{e}\mathit{t}\mathit{I}...$$

Commented by aliesam last updated on 29/Oct/19

$$thankyousomuchsir.godblessyou$$$$$$

Commented by Tanmay chaudhury last updated on 29/Oct/19

$$mostwelcomesir$$

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