Question

Evaluate : $\int \frac{dx}{3si{n}^{2}x+sinxcosx+1}$

• A. $-\frac{2}{\sqrt{15}}{\tan }^{-1}\left(\frac{2(\cot x-2)}{\sqrt{15}}\right)$
• B. $\frac{2}{\sqrt{3}}ta{n}^{-1}\left(\frac{3tanx-1}{\sqrt{1}5}\right)+c$
• C. $\frac{2}{\sqrt{3}}ta{n}^{-1}\left(\frac{3tanx+1}{\sqrt{1}5}\right)+c$
• D. $\frac{2}{\sqrt{15}}{\tan }^{-1}\left(\frac{2(\cot x-2)}{\sqrt{15}}\right)$

Answer 1

Answer: (D)

$\frac{2}{\sqrt{15}}{\tan }^{-1}\left(\frac{2(\cot x-2)}{\sqrt{15}}\right)$

Let $I=\int \frac{dx}{3{\sin }^{2}x+\sin x\cos x+1}$
Multiply numerator and denominator by ${\csc }^{2}x$
$I=\int \frac{{\csc }^{2}x}{\cot x+{\csc }^{2}x+3}dx=\int \frac{{\csc }^{2}x}{{\cot }^{2}x+\cot x+3}dx$
Substitute $t=\cot x\Rightarrow dt=-{\csc }^{2}x$
$I=-\int \frac{1}{t^2+t+4}dt=-\int \frac{1}{{(t-2)}^2+\frac{15}{4}}$
$=-\frac{2}{\sqrt{15}}{\tan }^{-1}\left(\frac{2(t-2)}{\sqrt{15}}\right)=-\frac{2}{\sqrt{15}}{\tan }^{-1}\left(\frac{2(\cot x-2)}{\sqrt{15}}\right)$