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If int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)d...

If `int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx=(1)/(12)tan^(-1)(3 tanx)+C`, then the value of ab , is

A

`41`

B

`12`

C

`39`

D

`36`

Text Solution

Verified by Experts

The correct Answer is:
B
We have , `I= int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx`
`rArr I= int(sec^(2)x)/(b^(2)+a^(2)tan^(2)x)dx`
` rArr I=(1)/(a)int(1)/(b^(2)+(atanx)^(2))d(atanx)`
`=(1)/(ab) tan^(-1)((a)/(b)tanx)+C`
`:. ab = 12 and (a)/(b) = 3 rArr a^(2)= 36 rArr a = +-6`
`:. ab = 12 rArr b =+-2`.
Thus , we have
`a sin x +b cos x = +- (6 sin x+2cosx)` We know that
`-sqrt(a^(2)+b^(2))le a sin x +b cos x le sqrt(a^(2)+b^(2))` for all x
` :. - sqrt(40)le +- 6 sin x +- 2 cos x le sqrt(40)` for allx.
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OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Chapter Test
  1. If int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx=(1)/(12)tan^(-1)(3 tanx)+C, ...

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  2. The integral int (2x-3)/(x^2+x+1)^2 .dx is equal to

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  3. If int(xtan^(-1)x)/sqrt(1+x^(2))dx = sqrt(1+x^(2))f(x) + A " ln "sqrt(...

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  4. "If " int xlog(1+1//x)dx=f(x)log(x+1)+g(x)x^(2)+Ax+C, then

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  5. If int(xe^(x))/(sqrt(1+e^(x)))dx=f(x)sqrt(1+e^(x))-2logg(x)+C, then

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  6. The value of int (cos^3x+cos^5)/(sin^2x+sin^4x)dx

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  7. If int(dx)/((x^(2)+1)(x^(2)+4))=k tan^(-1) x + l tan^(-1) . (x)/(2) +C...

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  8. If int log(sqrt(1-x)+sqrt(1+x))dx=xf(x)+Ax+Bsin^(-1)x+C, then

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  9. If int(x^(5))/(sqrt(1+x^(3)))dx is equal to

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  10. The value of : inte^(secx).sec^(3)x(sin^(2)x+cosx+sinx+sinxcosx)dx i...

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  11. If int(2x^(2)+3)/((x^(2)-1)(x^(2)+4))dx=aln((x-1)/(x+1))+btan^(-1).(x)...

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  12. Let f(x)=(x)/((1+x^(n))^(1//n)) for n ge 2 and g(x)=underset("n times"...

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  13. The value of int ((ax^(2)-b)dx)/(xsqrt(c^(2)x^(2)-(ax^(2)+b)^(2))) is ...

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  14. Evalaute: inte^(x)(1+nx^(n-1)-x^(2n))/((1-x^(n))sqrt(1-x^(2n))dx

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  15. int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2)))dx

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  16. int(x^(3))/((1+x^(2))^(1//3))dx is equal to

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  17. int sinx/sin(x-alpha)dx=Ax+B log (sin(x-alpha))+C then find out (A ,B)

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  18. What is int (x^(2) +1)/(x^(4) - x^(2) + 1) dx equal to ?

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  19. Evaluate: int(x-1)/((x+1)sqrt(x^3+x^2+x))dx

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  20. int(1+x^(2))/(xsqrt(1+x^(4)))dx is equal to

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  21. int(1+x^(4))/((1-x^(4))^(3//2))dx is equal to

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