dt-5cos-t-6sin-t-

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

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

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

I = \int \frac{d t}{5 \cos t + 6 \sin t}, x = \tan \frac{t}{2}

= \int \frac{2}{5 (\frac{1 - x^{2}}{1 + x^{2}}) + 6 (\frac{2 x}{1 + x^{2}})} \cdot \frac{d x}{1 + x^{2}}

= \int \frac{2 d x}{5 + 12 x - 5 x^{2}} = \int \frac{2 d x}{\frac{61}{5} - 5 {(x - \frac{6}{5})}^{2}}

= \frac{2}{\sqrt{61}} argtanh (\frac{5 x - 6}{\sqrt{61}}) + C

= \frac{1}{\sqrt{61}} \ln ∣ \frac{5 x + \sqrt{61} - 6}{5 x - \sqrt{61} - 6} ∣ + C

= \frac{1}{\sqrt{61}} \ln ∣ \frac{5 \tan (\frac{t}{2}) + \sqrt{61} - 6}{5 \tan (\frac{t}{2}) - \sqrt{61} - 6} ∣ + C

Commented by cortano1 last updated on 02/Sep/22

ok

Answered by blackmamba last updated on 01/Sep/22

\Leftrightarrow 5 \cos t + 6 \sin t = \sqrt{61} (\frac{5}{\sqrt{61}} \cos t + \frac{6}{\sqrt{61}} \sin t)

= \sqrt{61} \sin (α + t)

I = \int \frac{d t}{\sqrt{61} \sin (α + t)}

I = \frac{1}{\sqrt{61}} \ln ∣ \frac{1 - \cos (α + t)}{\sin (α + t)} ∣ + c

I = \frac{1}{\sqrt{61}} \ln ∣ \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} ∣ + c

I = \frac{1}{\sqrt{61}} \ln ∣ \frac{\sqrt{61} - 6 \cos t + 5 \sin t}{5 \cos t + 6 \sin t} ∣ + c

Commented by cortano1 last updated on 02/Sep/22

ok