Mathematics - Indefinite Integration Question with Solution | TestHub

We have:

$$I=\int e^{\sin x}\left(\frac{x{\cos }^{3}x-\sin x}{{\cos }^{2}x}\right)dx$$

This can be rewritten as:

$$=\int x{e}^{\sin x}\cos xdx-\int e^{\sin x}(\sec x\tan x)dx$$

Applying integration by parts, we get:

$$=\left[x{e}^{\sin x}-\int e^{\sin x}dx\right]-\left[e^{\sin x}\sec x-\int e^{\sin x}dx\right]$$

Simplifying the expression, the final result is:

$$=e^{\sin x}(x-\sec x)+C$$