Question

If $\int (2-3si{n}^{2}x)\sqrt{secx}dx=2f\left(x\right)\sqrt{g\left(x\right)}+c$ and f(x) is non constant function then

• A. $f^2\left(x\right)+g^2\left(x\right)=1$
• B. $f^2\left(x\right)-g^2\left(x\right)=1$
• C. $f\left(x\right)+g\left(x\right)=1$
• D. $f\left(x\right)+g\left(x\right)$

Answer 1

The correct option is A $f^2\left(x\right)+g^2\left(x\right)=1$

$\int \frac{(2-3si{n}^{2}x)}{\sqrt{cosx}}dx=\int \frac{2co{s}^{2}x-si{n}^{2}x}{\sqrt{cosx}}dx=2\int \left(cosx\right)\sqrt{cosx}dx-\int \frac{si{n}^{2}x}{\sqrt{cosx}}dx=2sinx\sqrt{cosx}+c\Rightarrow f\left(x\right)=sinx,g\left(x\right)=cosx$