Let's do some calculus! (19)

Calculus Level 3

\[\begin{eqnarray} I_{1} & = & \displaystyle \int_{0}^{{\pi}/{2}} {\dfrac{\sin x - \cos x}{1 + \sin x \cos x}} \,dx \\ \\ I_{2} & = & \displaystyle \int_{0}^{2 \pi} {{\cos}^{6}x} \,dx \\ \\ I_{3} & = & \displaystyle \int_{{-\pi}/{2}}^{{\pi}/{2}} {{\sin}^{3}x} \,dx \\ \\ I_{4} & = & \displaystyle \int_{0}^{1} {\ln \left( \dfrac{1}{x} - 1 \right)} \,dx \end{eqnarray}\]

From the given anti-derivatives above, find the one whose value is not equal to \(0\). If the anti-derivative is \(I_{n}\) where \(n \in \{1,2,3,4\}\), submit your answer as \(n\).

Notation: \(\ln \left( \cdot \right)\) denotes the natural logarithm function.


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