I asked ChatGPT to solve this integral and it gave a wrong answer. Can you check it?

Theres a fundamental difference between these two integrals. For $\int \frac{\sin^3 x}{\cos^2 x} dx$, your approach is almost correct. Let me complete it: Rewrite $\sin^3 x = \sin x (1 - \cos^2 x)$: $$\int \frac{\sin x (1 - \cos^2 x)}{\cos^2 x} dx = \int \frac{\sin x}{\cos^2 x} dx - \int \sin x dx$$ Let $u = \cos x$, $du = -\sin x\,dx$: $$-\int \frac{du}{u^2} + \cos x = \frac{1}{u} + \cos x + C = \sec x + \cos x + C$$ Lets verify by differentiating: $$\frac{d}{dx}(\sec x + \cos x) = \sec x \tan x - \sin x$$ $$= \frac{\sin x}{\cos^2 x} - \sin x = \frac{\sin x - \sin x \cos^2 x}{\cos^2 x} = \frac{\sin x(1 - \cos^2 x)}{\cos^2 x} = \frac{\sin^3 x}{\cos^2 x}$$ So YOUR answer $\sec x + \cos x + C$ is correct. The AI was wrong. Trust your math, not the chatbot.