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Evaluatethefollowingintegrals.4.∫(a−b)x(x−a)(x−b)dx(a>b)Evaluate the following integrals.4. \int \frac{(a-b) x}{(x-a)(x-b)} d x (a>b) Evaluatethefollowingintegrals.4.∫(x−a)(x−b)(a−b)xdx(a>b)
Example10:Evaluate(i)∫1a2−x2dx(−a<x<a)(ii)∫1xx2−a2dx(x>aorx<−a)whereaispositive.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.Example10:Evaluate(i)∫a2−x21dx(−a<x<a)(ii)∫xx2−a21dx(x>aorx<−a)whereaispositive.
5.Evaluatethefollowingintegrals.(viii)∫emTan−1x(1+x2)dx5. Evaluate the following integrals.(viii) \int \frac{e^{m \operatorname{Tan}^{-1} x}}{\left(1+x^{2}\right)} d x 5.Evaluatethefollowingintegrals.(viii)∫(1+x2)emTan−1xdx
22.Inaculturebacteriaincreasesattherateproportionaltothenumberofbacteriapresent.Ifbacteriaare200initiallyandaredoubledin2hoursfindthenumberofbacteriapresentfourhourslater.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.22.Inaculturebacteriaincreasesattherateproportionaltothenumberofbacteriapresent.Ifbacteriaare200initiallyandaredoubledin2hoursfindthenumberofbacteriapresentfourhourslater.
Evaluatethefollowingintegrals.12.∫4+7x(1+x)2(2+3x)dxEvaluate the following integrals.12. \int \frac{4+7 x}{(1+x)^{2}(2+3 x)} d x Evaluatethefollowingintegrals.12.∫(1+x)2(2+3x)4+7xdx
Solvethefollowingdifferentialequations:13.sec2xtanydx+sec2ytanxdy=0Solve the following differential equations:13. \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0 Solvethefollowingdifferentialequations:13.sec2xtanydx+sec2ytanxdy=0