Question

Assertion :STATEMENT-1 : $\int e^x\sin xdx=\frac{e^x}{2}(\sin x-\cos x)+c.$ Reason: STATEMENT-2 : $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+c.$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

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

$\int e^x\sin xdx$
$=\frac{1}{2}\int e^x(\sin x+\cos x+\sin x-\cos x)dx$
$=\frac{1}{2}(\int e^x(\sin x+\cos x)dx-\int e^x(\cos x-\sin x\left)dx\right)$
$=\frac{1}{2}(e^x\sin x-e^x\cos x)+c$
$=\frac{1}{2}{e}^x(\sin x-\cos x)+c$
Hence, assertion is correct.
Consider, $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx$
$=\int e^{x}f\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)-\int e^x{f}^{\prime }\left(x\right)dx+\int e^x{f}^{\prime }\left(x\right)dx$
$=e^{x}f\left(x\right)+C$
Hence, reason is correct. Reason is the correct explanation for assertion.