Question

Evaluate

$\int \frac{dx}{2-{3sin}^{2}x-{4cos}^{2}x}$

Answer 1

We have,

$\int \frac{dx}{2-3{\sin }^{2}x-4{\cos }^{2}x}dx$

$\Rightarrow \int \frac{dx}{2-3{\sin }^{2}x-3{\cos }^{2}x-{\cos }^{2}x}dx$

$\Rightarrow \int \frac{dx}{2-3({\sin }^{2}x+{\cos }^{2}x)-{\cos }^{2}x}dx$

$\Rightarrow \int \frac{dx}{2-3-{\cos }^{2}x}$

$\Rightarrow \int \frac{dx}{-1-{\cos }^{2}x}$

$\Rightarrow -1\int \frac{dx}{1+{\cos }^{2}x}$

Now, using formula,

$\int \frac{1}{\sqrt{1+x^2}}dx=\log |x+\sqrt{1+x^2}|$

Then,

$-\int \frac{dx}{1+{\cos }^{2}x}=-\log |\cos x+\sqrt{1+{\cos }^{2}x}|+C$

Hence, this is the answer.