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Pattern - 11 of Indefinite Integration

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Pattern – 11

$\frac{1}{a\cos^2{x}+b\sin^2{x}+c}\ or$ $\ \frac{1}{a\cos^2{x\ }+b}\ or$ $\ \frac{1}{a\sin^2{x}+b}\ or$ $\ \frac{1}{a\cos^2{x}+b\sin^2{x}}$

$1)\ I=\ \int\frac{1}{3+2\sin^2{x}+5\cos^2{x}}\ dx$

$=\ \int\frac{\sec^2{x}}{3\sec^2{x}+2\tan^2{x}+5}\ dx$

$=\ \int\frac{\sec^2{x}}{3\ \left(1+\tan^2{x}\right)+2\tan^2{x}+5}\ dx$

$=\ \int\frac{\sec^2{x}}{5\tan^2{x}+8\ }\ dx$

Let, tan x = t

∴ $\sec^2{x} dx$ = dt

$\therefore\ \ I=\ \int\frac{1}{5t^2+8}\ dt$

$=\ \int\frac{1}{\left(\sqrt5t\right)^2+\ \left(2\ \sqrt2\right)^2}\ dt$

$=\ \frac{1}{2\sqrt2}\ \times\frac{1}{\sqrt5}\times\tan^{-1}{\frac{\sqrt5\ t}{2\ \sqrt2}}+c$

$=\ \frac{1}{2\ \sqrt{10}}\tan^{-1}{\left(\frac{\sqrt5\tan{x}}{2\ \sqrt2}\right)}+c$