Question

Let f(x) be a polynomial function of degree 2 satisfying $\int \frac{f\left(x\right)}{x^3-1}=ln\left|\frac{x^2+x+1}{x-1}\right|+\frac{2}{\sqrt{3}}ta{n}^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c,$ where c is indefinite integration constant.
Let $\int \frac{5+f\left(sinx\right)+f\left(cosx\right)}{sinx+cosx}dx=h\left(x\right)+\lambda ,$ where h(1) = - 1. The value of $ta{n}^{-1}\left[h\right(2\left)\right]+ta{n}^{-1}\left[h\right(3\left)\right]$ 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

The correct option is D $\frac{-3\pi }{4}$

Now
$I=\int \frac{5+f\left(sinx\right)+f\left(cosx\right)}{sinx+cosx}dx=\int \frac{5+si{n}^{2}x-sinx-3+co{s}^{2}x-cosx-3}{sinx+cosx}=-x+\lambda h\left(x\right)=-xhence,ta{n}^{-1}\left(h\right(2\left)\right)+ta{n}^{-1}\left(h\right(3\left)\right)=\frac{-3\pi }{4}$