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Question
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Options
cot x + cosec x
cot2 x
cot x
cosec x
Solution
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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