# A contradictory integral: $\int \sin x \cos x \,dx$

I've been thinking about integration lately, and I've come up with a question that I'm not sure how to address. Consider $$\int \sin x\cos x \, dx = - \int -\sin x \cos x \, dx$$ I started with the integral on the left hand side, which suggests a typical $u$-substitution. Let $u=\sin x$ then $du=\cos x dx$. So the integral evaluates to $$\int \sin x\cos x \, dx = \frac{\sin^2(x)}{2}$$ But the original integral also suggests an alternate substitution. Let $u=\cos x$ and then $du=-\sin x \, dx$. So now $$\int \sin x\cos x \, dx =- \int -\sin x \cos x \, dx= -\frac{\cos^2(x)}{2}$$ So now I have that the integral evaluates to two different functions. I've tried playing with some different trigonometric identities, but I haven't been able to show that this is true and I'm fairly certain I haven't had any success because the statement itself isn't true. What am I doing wrong? How do you evaluate $\int \sin x\cos x \, dx$?

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$$\frac12\sin^2x=-\frac12\cos^2x+\frac12$$