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Question Number 83691 by niroj last updated on 05/Mar/20

$$\mathrm{evaluate}:$$$$\int \frac{\mathbf{dx}}{\mathbf{a}\mathbf{sin}\mathbf{x}+\mathbf{b}\mathbf{cos}\mathbf{x}}$$

Commented by mathmax by abdo last updated on 05/Mar/20

$$weusechangementtan\left(\frac{x}{2}\right)=t\Rightarrow$$$$I=\int \frac{1}{a\frac{2t}{1+t^2}+b\frac{1-t^2}{1+t^2}}\times \frac{2dt}{1+t^2}=2\int \frac{dt}{2at+b-{bt}^2}$$$$=-2\int \frac{dt}{{bt}^2-2at-b}$$$${\mathrm{\Delta }}^{{}^{\prime }}=a^2+b^2>0\Rightarrow t_1=\frac{a+\sqrt{a^2+b^2}}{b}and{t}_2=\frac{a-\sqrt{a^2+b^2}}{b}\Rightarrow$$$$I=-2\int \frac{dt}{b(t-t_1)(t-t_2)}=\frac{-2}{b(t_1-t_2)}\int (\frac{1}{t-t_1}-\frac{1}{t-t_2})dt$$$$=\frac{-2}{b\times \frac{2\sqrt{a^2+b^2}}{b}}ln\mid \frac{t-t_1}{t-t_2}\mid +C=\frac{-1}{\sqrt{a^2+b^2}}ln\mid \frac{tan\left(\frac{x}{2}\right)-\frac{a+\sqrt{a^2+b^2}}{b}}{tan\left(\frac{x}{2}\right)-\frac{a-\sqrt{a^2+b^2}}{b}}\mid +C$$$$=\frac{-1}{\sqrt{a^2+b^2}}ln\mid \frac{btan\left(\frac{x}{2}\right)-a-\sqrt{a^2+b^2}}{btan\left(\frac{x}{2}\right)-a+\sqrt{a^2+b^2}}\mid +C$$

Commented by john santu last updated on 06/Mar/20

$$\mathrm{if}\mathrm{you}\mathrm{want}\mathrm{to}\mathrm{short}\mathrm{cut}$$$$\int \frac{\mathrm{dx}}{\mathrm{asin}\mathrm{x}+\mathrm{bcos}\mathrm{x}}=\frac{1}{\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}\ln \mid \frac{\mathrm{a}+\mathrm{b}+\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}{\mathrm{a}+\mathrm{b}-\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}\mid +\mathrm{c}$$

Answered by john santu last updated on 05/Mar/20

$$\mathrm{asin}\mathrm{x}+\mathrm{b}\cos \mathrm{x}=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}\cos (\mathrm{x}-\theta )$$$$\mathrm{where}\theta ={\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)$$$$\Rightarrow \int \frac{\mathrm{dx}}{\mathrm{k}\cos (\mathrm{x}-\theta )},\mathrm{with}\mathrm{k}=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$$$$\Rightarrow \frac{1}{\mathrm{k}}\int \sec (\mathrm{x}-\theta )\mathrm{dx}=$$$$\Rightarrow \frac{1}{\mathrm{k}}\ln \mid \sec (\mathrm{x}-\theta )+\tan (\mathrm{x}-\theta )\mid +\mathrm{c}=$$$$\frac{1}{\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}\ln \mid \sec (\mathrm{x}-{\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right))+\tan (\mathrm{x}-{\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right))\mid +\mathrm{c}$$$$$$

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