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Second Year Math Integration


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Class 9Class 10First YearSecond Year
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.

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

Evaluatethefollowingintegrals.4.(ab)x(xa)(xb)dx(a>b)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.
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)1a2x2dx(a<x<a)(ii)1xx2a2dx(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.

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

5.Evaluatethefollowingintegrals.(viii)emTan1x(1+x2)dx5. 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.
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.
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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.

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

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

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

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

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