First Exam

1. $${ \int x \cos (x^{2} + 1) \, d x }$$
2. $${ \int x^{2} (x^{3} - 3)^{8} \, d x }$$
3. $${ \int \frac{1}{\sqrt{4-x}} \, d x }$$
4. $${ \int \sin \theta \cos^{6} \theta \, d \theta }$$
5. $${ \int_{1}^{4} x e^{5x} \, d x }$$
6. $${ \int x^{3} \ln x \, d x }$$
7. $${ \int \frac{x+1}{6x + x^{2}} \, d x }$$
8. $${ \int \frac{x^{4} + 3 x^{3} + 2 x^{2} + 1}{x^{2} + 3 x + 2} \, dx }$$
9. $${ \int \frac{1}{x^{2} + 4 x + 5} \, d x }$$
10. $${ \int_{1}^{5} \frac{dx}{x^{3/2}} }$$
11. $${ \int (x+ \sin x)^{3} (1 + \cos x) \, dx }$$
12. $${ \int \sin 3 x \, \sin 5 x \, dx }$$
    Hint: $\sin A \, \sin B = \frac{1}{2} [\cos (A-B) - \cos (A+B)]$
13. $${ \int \frac{x^{3}}{\sqrt{x^{2} + 9}} d x }$$
    Hint: let $x=3 \tan \theta$ .