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Ifint(3e^x-5e^(-x))/(4e^x+5e^(-x))dx=a x...

`Ifint(3e^x-5e^(-x))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n` (a) `a=-1/8,b=7/8` (b) `a=1/8,b=7/8` (c) `a=-1/8,b=-7/8` (d) `a=1/8,b=-7/8""`

A

`a= -(1)/(8), b=(7)/(8)`

B

`a= (1)/(8), b=(7)/(8)`

C

`a= -(1)/(8), b= -(7)/(8)`

D

`a= (1)/(8), b= -(7)/(8)`

Text Solution

Verified by Experts

The correct Answer is:
A
`int(3e^(x)-5e^(-x))/(4e^(x)+5e^(-x))dx=ax+b " In " (4e^(x)+5e^(-x))+C`
Differentiating both sides, we get
`(3e^(x)-5e^(-x))/(4e^(x)+5e^(-x))=a+b ((4e^(x)-5e^(-x)))/(4e^(x)+5e^(-x))`
or `3e^(x)-5e^(-x)=a(4e^(x)+5e^(-x))=b(4e^(x)-5e^(-x))`
Comparing the coefficients of like terms on both sides, we get
`3=4(a+b),-5=5a-5b" or " a= -(1)/(8), b=(7)/(8)`
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CENGAGE ENGLISH-INDEFINITE INTEGRATION-EXERCISES (Single Correct Answer Type)
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  2. If int(dx)/((x+2)(x^(2)+1)) = alog|1+x^(2)|+btan^(-1)x+ 1/5log|x+2|+C,...

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  3. Ifint(3e^x-5e^(-x))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n ...

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  5. int(x^(9))/((4x^(2)+ 1)^(6))dx is equal to

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  6. If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b , th...

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  7. The value of the integral int(x^2+x)(x^(-8)+2x^(-9))^(1/(10))dx is 5/(...

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  8. int(x^3dx)/(sqrt(1+x^2))i se q u a lto 1/3sqrt(1+x^2)(2+x^2)+C 1/3...

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  9. If I=int (dx)/((a^(2)-b^(2)x^(2))^(3//2)), then I equals

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  10. int((x^4-1)dx)/(x^2sqrt(x^4+x^2+1))

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  11. int(sqrt(x^2+1))/(x^4)dx=

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  12. int (1+2x^(6))/((1-x^(6))^(3//2))dx is equal to

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  13. The integral int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx is equal to-

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  14. If In=int( lnx)^n dx then In+nI(n-1)

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  15. int e^x {f(x)-f'(x)}dx= phi(x), then int e^x f(x) dx is

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  16. For x >1,int sin^(- 1)((2x)/(1+x^2))dx is equal to

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  17. If int x((ln(x+sqrt(1+x^2)))/sqrt(1+x^2)) dx=asqrt(1+x^2)ln(x+sqrt(1+x...

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  18. Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C , then f(x)=1/2x^2 (b) ...

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  19. If I=inte^(-x)log(e^x+1)dx ,t h e nIe q u a l a+(e^(-x)+1)log(e^x+1)...

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  20. "If " int x e^(x) cosx dx=ae^(x)(b(1-x)sinx+cx cosx)+d, then

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