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Question Number 175531 by cortano1 last updated on 01/Sep/22

$$\int \frac{dt}{5\cos t+6\sin t}=?$$

Answered by Ar Brandon last updated on 01/Sep/22

$$I=\int \frac{dt}{5\cos t+6\sin t},x=\tan \frac{t}{2}$$$$=\int \frac{2}{5\left(\frac{1-x^2}{1+x^2}\right)+6\left(\frac{2x}{1+x^2}\right)}\cdot \frac{dx}{1+x^2}$$$$=\int \frac{2dx}{5+12x-5x^2}=\int \frac{2dx}{\frac{61}{5}-5{(x-\frac{6}{5})}^2}$$$$=\frac{2}{\sqrt{61}}\mathrm{argtanh}\left(\frac{5x-6}{\sqrt{61}}\right)+C$$$$=\frac{1}{\sqrt{61}}\ln \mid \frac{5x+\sqrt{61}-6}{5x-\sqrt{61}-6}\mid +C$$$$=\frac{1}{\sqrt{61}}\ln \mid \frac{5\tan \left(\frac{t}{2}\right)+\sqrt{61}-6}{5\tan \left(\frac{t}{2}\right)-\sqrt{61}-6}\mid +C$$

Commented by cortano1 last updated on 02/Sep/22

$$\mathrm{ok}$$

Answered by blackmamba last updated on 01/Sep/22

$$\iff 5\cos t+6\sin t=\sqrt{61}(\frac{5}{\sqrt{61}}\cos t+\frac{6}{\sqrt{61}}\sin t)$$$$=\sqrt{61}\sin (\alpha +t)$$$$I=\int \frac{dt}{\sqrt{61}\sin (\alpha +t)}$$$$I=\frac{1}{\sqrt{61}}\ln \mid \frac{1-\cos (\alpha +t)}{\sin (\alpha +t)}\mid +c$$$$I=\frac{1}{\sqrt{61}}\ln \mid \frac{1-(\frac{6}{\sqrt{61}}\cos t-\frac{5}{\sqrt{61}}\sin t)}{\frac{5}{\sqrt{61}}\cos t+\frac{6}{\sqrt{61}}\sin t}\mid +c$$$$I=\frac{1}{\sqrt{61}}\ln \mid \frac{\sqrt{61}-6\cos t+5\sin t}{5\cos t+6\sin t}\mid +c$$

Commented by cortano1 last updated on 02/Sep/22

$$\mathrm{ok}$$

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