Trigonometric Substitutions: 

Evaluate the following integrals:

  1. \( \displaystyle{\int \frac{x}{\sqrt{4-x^2}} \ \mathrm{d}x }\)   solution

  2. \( \displaystyle{\int \frac{x^2}{\sqrt{9-x^2}} \ \mathrm{d}x}\)  solution

  3. \( \displaystyle{\int \frac{\sqrt{x^2-4}}{x} \ \mathrm{d}x}\)  solution

  4. \(\displaystyle{\displaystyle{ \int_0^1 x^3 \sqrt{1-x^2} \ \mathrm{d}x}}\)   solution

  5. \( \displaystyle{\int \frac{1}{\sqrt{4+x^2}} \ \mathrm{d}x }\)   solution

  6. \( \displaystyle{\int \frac{\sqrt{x^2+ 1}}{x} \ \mathrm{d}x}\)   solution

  7. \( \displaystyle{\int \frac{x^3}{\sqrt{x^2+25}} \ \mathrm{d}x}\)  solution

  8. \(\displaystyle{\int \frac{x}{\sqrt{x^2-25}} \ \mathrm{d}x }\)   solution

  9. \( \displaystyle{\int \frac{\sqrt{x^2-25}}{x^3} \ \mathrm{d}x}\)  solution

  10. \( \displaystyle{\int_0^5 \frac{dt}{\sqrt{25+t^2}} \ \mathrm{d}t}\)  solution

  11. \(\displaystyle{\int \sqrt{1-4x^2} \ \mathrm{d}x }\)   solution

  12. \(\displaystyle{ \displaystyle{\int_{\sqrt{2}}^2 \frac{1}{x^3\sqrt{x^2-1}} \ \mathrm{d}x} }\)   solution

  13. \(\displaystyle{\displaystyle{ \int_0^{\frac{3\sqrt{3}}{2}} \frac{x^3}{(4x^2+9)^{3/2}} \ \mathrm{d}x} }\)   solution