1. Each of these integrals are improper. Identify what makes each one improper, and decide if each integral converges or diverges. If it converges, compute its value. \[ \int\limits_{1}^{\infty} \frac{\mathrm{d}x}{x^5} \qquad \int\limits_{1}^{\infty} \frac{\mathrm{d}x}{\sqrt{x}} \qquad \int\limits_{0}^{\infty} \mathrm{e}^{-x} \,\mathrm{d}x \] \[ \int\limits_{0}^{1} \frac{\mathrm{d}x}{x^5} \qquad \int\limits_{0}^{1} \frac{\mathrm{d}x}{\sqrt{x}} \qquad \int\limits_{-\infty}^{0} \mathrm{e}^{-x} \,\mathrm{d}x \] \[ \int\limits_{0}^{1} \ln(x) \,\mathrm{d}x \qquad \int\limits_{0}^{3} \frac{1}{x^2-1} \,\mathrm{d}x \qquad \int\limits_{0}^{\pi/2} \sec(x) \,\mathrm{d}x \]
2. Determine whether each of the following integrals converges or diverges. \[ \int\limits_{1}^{\infty} \frac{1+\cos^2(x)}{\sqrt[3]{x}} \,\mathrm{d}x \qquad \int\limits_{13}^{\infty} \frac{x+43}{x^2-144} \,\mathrm{d}x \qquad \int\limits_{0}^{\infty} \frac{\arctan(5x)}{x^2+3x+7} \,\mathrm{d}x \] \[ \int\limits_{1}^{\infty} \frac{1}{x}\mathrm{e}^{-x} \,\mathrm{d}x \qquad \int\limits_{0}^{3} \frac{x+1}{x-1} \,\mathrm{d}x \qquad \int\limits_{1}^{\infty} \frac{x+11}{\sqrt{x^5-x^2}} \,\mathrm{d}x \qquad \int\limits_{1}^{\infty} \frac{x-2}{x^3+125} \,\mathrm{d}x \]