1003107804

Four girls evaluated the integral $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x$ on $\mathbb{R}$. Ann started to integrate by parts like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=\sin^2⁡x-\int\cos x\cdot\sin x\,\mathrm{d}x$. Beth started to integrate by parts like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=-\cos^2 x-\int\sin x\cdot\cos x\,\mathrm{d}x$. Claire used substitution $a=\sin ⁡x$ like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=\int a\,\mathrm{d}a$. Diana integrated $\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=-\cos x\cdot\sin⁡ x+c$, $c\in\mathbb{R}$. Which of the girls made a mistake?