If $u=\int e^{a x} \sin b x d x$ and $v=\int e^{a x} \cos b x d x$, then $\left(u^2+v^2\right)\left(a^2+b^2\right)=$
Answer & explanation
Correct answer: option 2
We have,
$u =\int e^{a x} \sin b x d x$ and $v=\int e^{a x} \cos b x d x$
$\Rightarrow u =\frac{e^{a x}}{a^2+b^2}(a \sin b x-b \cos b x)$
and, $v=\frac{e^{a x}}{a^2+b^2}(a \cos b x+b \sin b x)$
∴ $\left(u^2+v^2\right)\left(a^2+b^2\right)^2=e^{2 a x}\left(a^2+b^2\right)$
$\Rightarrow \left(u^2+v^2\right)\left(a^2+b^2\right)=e^{2 a x}$