If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.

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If ∫(cot x – cosec2x)exdx = ex f(x) + c then f(x) will be cot x. Explanation: ∫(cot x – cosec2x)ex dx = ex f(x) + c Then, ∫(cot x – cosec2x)ex dx = ∫ex cot x dx – ∫ex cosec2 x dx On integrating by parts = `cot x int e^x dx - int d/(dx) cot x int e^x dx - int e^x "cosec"^2 dx + c` = ex cot x + ∫ex cosec2x dx – ∫ex cosec2 dx + c = ex cot x + c Hence, f(x) = cot x.

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