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##### Evaluate the following definite integrals.1. \int_{1}^{2}\left(x^{2}+1\right) d x 2. \int_{-1}^{1}\left(x^{1 / 3}+1\right) d x 3. \int_{-2}^{0} \frac{1}{(2 x-1)^{2}} d x 4. \int_{-6}^{2} \sqrt{3-x} d x 5. \int^{\sqrt{ }} \sqrt{(2 t 1)} d t 6. \int_{2}^{\sqrt{5}} x \sqrt{x^{2}-1} d x 7. \int_{1}^{2} \frac{x}{x^{2}+2} d x 8. \int_{2}^{3}\left(x-\frac{1}{x}\right)^{2} d x 9. \int_{-1}^{1}\left(x+\frac{1}{2}\right) \sqrt{x^{2}+x+1} d x 10. \int_{0}^{3} \frac{d x}{x^{2}+9} 11. \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos t d t 12. \int_{1}^{2}\left(x+\frac{1}{x}\right)^{\frac{1}{2}}\left(1-\frac{1}{x^{2}}\right) d x 13. \int_{1}^{2} \operatorname{In} x d x 14. \int_{0}^{2}\left(e^{\frac{x}{2}}-e^{-\frac{x}{2}}\right) d x 15. \int_{0}^{\frac{\pi}{4}} \frac{\cos \theta+\sin \theta}{2 \cos ^{2} \theta} d \theta 16. \int_{0}^{\frac{\pi}{6}} \cos ^{3} \theta d \theta 17. \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos ^{2} \theta \cot ^{2} \theta d \theta 18. \int_{0}^{\frac{\pi}{4}} \cos ^{4} t d t

Evaluate the following integrals.4. \int \frac{(a-b) x}{(x-a)(x-b)} d x (a>b)

Example 10: Evaluate(i) \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x (-a<x<a) (ii) \int \frac{1}{x \sqrt{x^{2}-a^{2}}} d x(x>a or x<-a) where a is positive.

5. Evaluate the following integrals.(viii) \int \frac{e^{m \operatorname{Tan}^{-1} x}}{\left(1+x^{2}\right)} d x

22. In a culture bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 200 initially and are doubled in 2 hours find the number of bacteria present four hours later.

Evaluate the following integrals.12. \int \frac{4+7 x}{(1+x)^{2}(2+3 x)} d x

Solve the following differential equations:13. \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0