Search any question & find its solution
Question:
Answered & Verified by Expert
Integrate the function
$e^{2 x} \sin x$
$e^{2 x} \sin x$
Solution:
1714 Upvotes
Verified Answer
Let $I=\int e^{2 x} \sin x d x$
$\begin{aligned}
&=e^{2 x}(-\cos x)-\int 2 e^{2 x}(-\cos x) d x \\
&=-e^{2 x} \cos x+2\left[e^{2 x}(\sin x)-\int 2 e^{2 x}(\sin x) d x\right] \\
&=e^{2 x}(2 \sin x-\cos x)-4 I \\
&\Rightarrow I=\frac{e^{2 x}}{5}(2 \sin x-\cos x)+C
\end{aligned}$
$\begin{aligned}
&=e^{2 x}(-\cos x)-\int 2 e^{2 x}(-\cos x) d x \\
&=-e^{2 x} \cos x+2\left[e^{2 x}(\sin x)-\int 2 e^{2 x}(\sin x) d x\right] \\
&=e^{2 x}(2 \sin x-\cos x)-4 I \\
&\Rightarrow I=\frac{e^{2 x}}{5}(2 \sin x-\cos x)+C
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.