First slide

Methods of integration

Question

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 $e^{x} f (x)$

Moderate

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

Solution

Let $I = \int e^{x} \sin x d x = - e^{x} \cos x + \int e^{x} \cos x d x$
$\begin{array}{l} = - e^{x} \cos x + e^{x} \sin x - \int e^{x} \sin x d x + \\ 2 I = e^{x} (\sin x - \cos x) + C \\ \Rightarrow I = \frac{e^{x}}{2} (\sin x - \cos x) + C \end{array}$
Statement 2: is also true but is not correct explanation of
statement 1.

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