CBSE Class 12 · 2022 · Indefinite IntegralsShort Answer

Find

\int e^{x} sin 2 x d x

Answer: $\frac{e ^{x}}{5} (sin 2 x - 2 cos 2 x) + C$

Solution

Let

I = \int e^{x} sin 2 x d x

. Integrate by parts twice (taking

e^{x}

as the second function):

I = e^{x} sin 2 x - 2 \int e^{x} cos 2 x d x .

\int e^{x} cos 2 x d x = e^{x} cos 2 x + 2 \int e^{x} sin 2 x d x = e^{x} cos 2 x + 2 I .

Substituting back,

I = e^{x} sin 2 x - 2 e^{x} cos 2 x - 4 I \Rightarrow 5 I = e^{x} (sin 2 x - 2 cos 2 x) .

\frac{e ^{x}}{5} (sin 2 x - 2 cos 2 x) + C

CBSE Class 12 Board · 2022 · Subjective