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Question Number 43626 by peter frank last updated on 12/Sep/18

$$\int \frac{x\cos x+1}{\sqrt{2x^3{e}^{\sin x}+x^2}}dx=$$

Answered by MJS last updated on 15/Sep/18

$$\int \frac{1+x\cos x}{\sqrt{2x^3{\mathrm{e}}^{\sin x}+x^2}}dx=\int \frac{1+x\cos x}{x\sqrt{2x{\mathrm{e}}^{\sin x}+1}}=$$$$[t=2x{\mathrm{e}}^{\sin x}+1\to dx=\frac{dt}{{2\mathrm{e}}^{\sin x}(1+x\cos x)}]$$$$=\int \frac{dt}{(t-1)\sqrt{t}}=$$$$[u=\sqrt{t}\to dt=2\sqrt{u}du]$$$$=2\int \frac{du}{u^2-1}=2\int \frac{du}{(u-1)(u+1)}=\int \frac{du}{u-1}-\int \frac{du}{u+1}=$$$$=\ln (u-1)-\ln (u+1)=\ln \frac{u-1}{u+1}=$$$$=\ln \frac{\sqrt{t}-1}{\sqrt{t}+1}=\ln \mid \frac{\sqrt{2x{\mathrm{e}}^{\sin x}+1}-1}{\sqrt{2x{\mathrm{e}}^{\sin x}+1}+1}\mid +C$$

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