# Second Year Math Integration

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

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

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

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

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

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