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Calculus Level 5

The equation of the curve \(y\)=\(f(x)\).The tangents at :-

(1,\(f(1)\)),(2,\(f(2)\)) ,and(3,\(f(3)\)) makes angle \(\dfrac{\pi}{6}\),\(\dfrac{\pi}{3}\) and \(\dfrac{\pi}{4}\) respectively with the positive direction of \(x\)- axis.Then the value of

\(\displaystyle\int_{2}^{3}\)\(f'(x)\)\(f"(x)\)\(dx\)+ \(\displaystyle\int_{1}^{3}\)\(f"(x)\)\(dx\) can be expressed as

\(\dfrac{-1}{a}\) find the value of \(a\).

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