NJC Integration Techniques Tutorial Q2

Use the given substitution to find the exact numerical values of the definite integrals.

5. \(\displaystyle\displaystyle \displaystyle\int_{0}^{\sqrt{3}}{\frac{{{x}^{2}}}{{{\left( 1+{{x}^{2}} \right)}^{2}}}\mathrm{d}x}\) \(\qquad\) \(x=\tan \theta \)
6. \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\sqrt{3}}{2}}{\sqrt{4-4{{x}^{2}}}\mathrm{d}x}\) \(\qquad\) \(x=\cos \theta \)

9. \(\displaystyle\displaystyle \displaystyle\int_{0}^{\infty }{\frac{{{x}^{2}}}{1+{{x}^{6}}}\mathrm{d}x}\) \(\qquad\) \(\qquad\) \(u={{x}^{3}}\)
10. \(\displaystyle\displaystyle \displaystyle\int_{0}^{a}{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}\mathrm{d}x}\) \(\qquad\) \(x=a\sin \theta \), \(a>0\)