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Statement 1: $\int e^{x} \sin x d x = \frac{e^{x}}{2} (\sin x - \cos x)$
$+ C$
Statement 2: If the integrand is the form of $e^{x} (f (x) + f^{'} (x)$
then the integrals in $e^{x} f (x)$

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STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation forSTATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

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detailed solution

Correct option is B

Let I=∫exsin⁡xdx=−excos⁡x+∫excos⁡xdx=−excos⁡x+exsin⁡x−∫exsin⁡xdx+ 2I=ex(sin⁡x−cos⁡x)+C⇒ I=ex2(sin⁡x−cos⁡x)+CStatement 2: is also true but is not correct explanation ofstatement 1.

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Statement 1: ∫exsin⁡xdx=ex2(sin⁡x−cos⁡x)+CStatement 2: If the integrand is the form of exf(x)+f′(x) then the integrals in exf(x)

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Statement 1: $\int e^{x} \sin x d x = \frac{e^{x}}{2} (\sin x - \cos x)$
$+ C$
Statement 2: If the integrand is the form of $e^{x} (f (x) + f^{'} (x)$
then the integrals in $e^{x} f (x)$