Trigonometric Substitutions: 

General rule (watch the video)

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

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

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

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

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

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

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