Question

Let $f(x)$ be a polynomial function of degree 2 satisfying
$\int \frac{f(x)}{x^3-1}dx=\ln \left|\frac{x^2+x+1}{x-1}\right|+\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+C,$
where $C$ is indefinite integration constant.
Let $\int \frac{5+f(\sin x)+f(\cos x)}{\sin x+\cos x}dx=h(x)+\lambda$, where $h(1)=-1$. The value of ${\tan }^{-1}(h(2))+{\tan }^{-1}(h(3))$ is equal to (where $\lambda$ is indefinite integration constant.)

• A. $\frac{\pi }{4}$
• B. $\frac{-\pi }{4}$
• C. $\frac{3\pi }{4}$
• D. $\frac{-3\pi }{4}$

Answer 1

Answer: (D)

$I=\int \frac{5+f(\sin x)+f(\cos x)}{\sin x+\cos x}dx=\int \frac{5+{\sin }^{2}x-\sin x-3+{\cos }^{2}x-\cos x-3}{\sin x+\cos x}dx$
$=\int -dx=-x+\lambda$
$\therefore h(x)=-x(\text{ since }h(1)=-1)$
Hence ${\tan }^{-1}(h(2))+{\tan }^{-1}(h(3))={\tan }^{-1}(-2)+{\tan }^{-1}(-3)=\frac{-3\pi }{4}$