# ∫ E 2 X Sin X D X - Mathematics

Sum
$\int e^{2x} \sin x\ dx$

#### Solution

$\text{ Let I }= \int e^{2x} \text{ sin x dx }$
\text{Considering sin  x  as first function and \text{ e}^{2x}    \text{ as second function}
$I = \sin x\frac{e^{2x}}{2} - \int \cos x\frac{e^{2x}}{2}dx$
$\Rightarrow I = \text{ sin x}\frac{e^{2x}}{2} - \frac{1}{2}\int \text{ cos x e }^{2x} \text{ dx }$
$\Rightarrow I = \frac{\text{ sin x e}^{2x}}{2} - \frac{1}{2}\left[ \cos x\frac{e^{2x}}{2} - \int\left( - \sin x \right)\frac{e^{2x}}{2}dx \right]$
$\Rightarrow I = \frac{\text{ sin x e}^{2x}}{2} - \frac{\text{ cos x e}^{2x}}{4} - \frac{1}{2}\int\frac{e^{2x} \sin x}{2}dx$
$I = \frac{e^{2x} \left( 2 \sin x - \cos x \right)}{4} - \frac{I}{4}$
$\Rightarrow 5I = e^{2x} \left( 2 \sin x - \cos x \right)$
$\Rightarrow I = \frac{e^{2x} \left( 2 \sin x - \cos x \right)}{5} + C$

Concept: Integrals of Some Particular Functions
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.27 | Q 6 | Page 149