Trigonometric Substitution

  1. Spivak

    Evaluate the following integrals. Only do as many as it takes you to feel confident that you can recognize the correct substitution. \[ \int \frac{\mathrm{d}x}{1+x^2} \qquad \int \frac{\mathrm{d}x}{1-x^2} \qquad \int \frac{\mathrm{d}x}{\sqrt{1-x^2}} \] \[ \int \frac{\mathrm{d}x}{\sqrt{1+x^2}} \qquad \int \frac{\mathrm{d}x}{\sqrt{x^2-1}} \qquad \int \frac{\mathrm{d}x}{x\sqrt{x^2-1}} \] \[ \int \frac{\mathrm{d}x}{x\sqrt{1-x^2}} \qquad \int \frac{\mathrm{d}x}{x\sqrt{1+x^2}} \qquad \int x^3\sqrt{1-x^2}\,\mathrm{d}x \] \[ \int \sqrt{1-x^2}\,\mathrm{d}x \qquad \int \sqrt{1+x^2}\,\mathrm{d}x \qquad \int \sqrt{x^2-1}\,\mathrm{d}x \]