Question

$\int e^x\left(\sin x+2\cos x\right)\sin xdx=$

• A. $e^x\cos x+c$
• B. $e^x\sin x+c$
• C. $e^x{\sin }^{2}x+c$
• D. $e^x\sin 2x+c$
• E. $e^x(\cos x+\sin x)+c$

Answer 1

The correct option is C

$e^x{\sin }^{2}x+c$

Explanation for the correct Option:

Integrating using substitution:

Given

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

Let's take $$\begin{gathered} f\left(x\right)=\sin x^2 \\ f\text{'}\left(x\right)=2\sin x\cos x \end{gathered}$$

Substituting the values, we get

$$\begin{gathered} =\int e^x\left(f\left(x\right)+f\text{'}\left(x\right)\right)dx \\ =e^{x}f\left(x\right)+c\left[\because \int e^x\left(f\left(x\right)+f\text{'}\left(x\right)\right)dx=e^{x}f\left(x\right)+c\right] \\ =e^x{\sin }^{2}x+c\left[\because f\left(x\right)={\sin }^{2}x\right] \end{gathered}$$

Hence, the correct answer is option (C).