1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f(x) be a polynomial function of degree 2 satisfying ∫ ( f ( x )/ x 3-1) dx = ln |( x 2+ x +1/ x -1)|+(2/√3) tan -1((2 x +1/√3))+ C , where C is indefinite integration constant. Let ∫ (5+f( sin x)+f( cos x)/ sin x+ cos x) d x=h(x)+λ, where h(1)=-1. The value of tan -1( h (2))+ tan -1( h (3)) is equal to (where λ is indefinite integration constant.)

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

Integrals

Solution:

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