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Geometric Interpretation of Def.Int as Limit of Sum

If the value ...

Question

If the value of $\int ({sin}^{n} 3 x + {cos}^{n} 3 x) d x = x + C,$ then the value of $n$ is
(where $C$ is constant of integration)

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Solution

Let $I = \int ({sin}^{n} 3 x + {cos}^{n} 3 x) d x$
$I = x + C$ is possible only when the integrant is $1 .$
i.e $I = \int 1 \cdot d x$
This is possible only when $n = 2$ so that ${sin}^{2} 3 x + {cos}^{2} 3 x = 1$

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