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Pattern - 12 of Indefinite Integration

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Pattern – 12

$\frac{A\sin{x}\times \cos{x}}{a\cos^2{x}+b\sin^2{x}+c}\ OR\ \ \frac{A\tan{x}}{a\tan^2{x}+b\sec^2{x}+c}$

$1)\ I=\ \int\frac{\tan{x}}{3\tan^2{x}-4\sec^2{x}+9\ }\ dx$

$=\ \int\frac{\sin{x}\times \cos{x}}{3\sin^2{x}-4+9\cos^2{x}}\ dx$

$Let, 3 \sin^2{x}+9\cos^2{x}-4=t$

∴ ( 6 sin x ∙ cos x – 18 cos x sin x )dx = dt

∴ -12 sin x ∙ cos x dx = dt

$\therefore\ \sin{x\ }\times \cos{x}\ dx=\ \frac{-1}{12}\ dt$

$\therefore\ \ I=\ \frac{-1}{12}\ \int\frac{1}{\ \ t\ \ }\ dt$

$=\ \frac{-1}{12}\ log\left|t\right|+c$

$=\ \frac{-1}{12}\ \log{\left|3\sin^2{x}+9\cos^2{x}-4\right|}+c$

$=\ \frac{-1}{12}\log{\left|6\cos^2{x}-1\right|}+c$