Question

The value of $\int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx}$ is

Answer 1

Find the value of $\int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx}$ .

Given,

The value is $\int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx}$.

The above value is in the form of

$\int e^x\left[f\left(x\right)+f^{\text{'}}\left(x\right)\right]dx=e^{x}f\left(x\right)+c$

So,

$$\begin{gathered} \int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx} \\ \int e^{2x}(2\sin 3x+3\cos 3x)dx \\ =\int d\left(e^{2x}\sin 3x\right) \\ =e^{2x}\sin 3x+c \end{gathered}$$

Hence, the required value of $\int {\mathrm{e}}^{2\mathrm{x}}(2\sin 3\mathrm{x}+3\cos 3\mathrm{x})\mathrm{dx}$is $e^{2x}\sin 3x+c$.