∫exsinx+cosxdx is equal to - exsin x + C excos x + C ex

SOLUTION

Whenever you come across an integral which has $e^x$ with some other function in its integrand, it is advised to check whether the integral is of the form of $\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx$. In that case, we can use the formula $\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx=e^{x}f\left(x\right)$

Here, we can see that cos x is the derivative of sin x, so the given integral is of the form $\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx$. Hence we can directly say that the solution of this would be $e^x\text{sin}x+C$.