Question

Write the value of $\int e^x(\sin x+\cos x)dx$.

Answer 1

The integral $\int e^x(\sin x+\cos x)dx$ is of the form $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx$, where $f\left(x\right)=\sin x$ and $f^{\prime }\left(x\right)=\cos x$

We know that $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+c$, where $c$ is a constant of integration.

$\therefore \int e^x(\sin x+\cos x)dx=e^x\sin x+c$.