Solve ∫ e^x*sin(e^x-2) wrt y

$$\int e ^ { x } \cdot \sin ( e ^ { x } - 2 ) d y$$

Evaluate

$$ye^{x}\sin(e^{x}-2)+С$$

Solution Steps

Find the integral of e^{x}\sin(e^{x}-2) using the table of common integrals rule \int a\mathrm{d}y=ay.

$$e^{x}\sin(e^{x}-2)y$$

If F\left(y\right) is an antiderivative of f\left(y\right), then the set of all antiderivatives of f\left(y\right) is given by F\left(y\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$e^{x}\sin(e^{x}-2)y+С$$

Differentiate w.r.t. x

$$ye^{x}\left(e^{x}\cos(e^{x}-2)+\sin(e^{x}-2)\right)$$