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int(3 sinx+ 2 cos x)/(3 cosx+2 sin x)dx=...

`int(3 sinx+ 2 cos x)/(3 cosx+2 sin x)dx= ax+b log |3cos x + 2 sin x| +C`, then (a ,b)

A

`a=(5)/(13), b =-(12)/(13)`

B

`a=(12)/(13),b =-(5)/(13)`

C

`a=(12)/(13),b=(5)/(13)`

D

`a=(-12)/(5),b=(-5)/(13)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `I= int(3 sin x+2 cos x)/(3 cos x+2sinx)dx`
Let `3 sin x +2 cos x= lambda (d)/(dx)(3cosx +2sinx)+mu(3 cos x +2 sinx)`
`rArr 3 sin x +2 cos x = lambda(-3 sin x +2 cos x )+mu (3 cos x+2 sin x)`
Comparing the coefficients sinx and cos x on both sides , we get
`-3lambda_2mu = 3 and 2 lambda +3 mu =2`
` rArr lambda =-5 //13 and = 12//13`
`:. I= int(-((5)/(13))(-3sin x +2 cos x)+((12)/(13))(3 cos x +2 sin x))/(3 cos x+ 2 sin x)dx`
`rArr I=(12)/(13)int1*dx - (5)/(13) int (1)/(3 cos x +2sin x)d(3cos x +2 sin x)`
`rArr I=(12)/(13) x-(5)/(13)log |3 cos x+2 sin x| +C`
Hence , a `=(12)/(13) and b = (-5)/(13)`
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  10. The value of inte^(secx)*sec^3x(sin^2x+cosx+sinx+sinxcosx)dx is

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  11. int(2x^(2)+3)/((x^(2)-1)(x^(2)+4))dx=alog((x+1)/(x-1))+b"tan"^(-1)(x)/...

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