Primitive function

Evaluate the following indefinite integral on \( \left(-\frac{\pi}2;\frac{\pi}2 \right) \). \[ \int \left(\frac1{\cos x}-\sin x\cdot\mathrm{tg}\,x\right)\,\mathrm{d}x \]

Choose an incorrect evaluation of the following indefinite integral on \( (0;\infty) \). \[ \int\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\,\mathrm{d}x \]

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?

Solve an indefinite integral $\int\frac{8x^7-30x^5}{x^8-5x^6+2}\,\mathrm{d}x$ on $(3;\infty)$.