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Let f(x)=int(dx)/(e^(x)+8e^(-x)+4e^(-3x)...

Let `f(x)=int(dx)/(e^(x)+8e^(-x)+4e^(-3x)),g(x)=int(dx)/(e^(3x)+8e^(x)+4e^(-x)).`
`int(f(x)-2g(x))dx`

A

`(1)/(2)log|(e^(x)+2e^(-x)-2)/(e^(x)+2e^(-x)+2)|+C`

B

`(1)/(4sqrt3)log|(e^(x)-2e^(-x)-2sqrt3)/(e^(x)+2e^(-x)+2sqrt3)|+C`

C

`(1)/(2sqrt3)tan^(-1)((e^(x)-2e^(-x))/(2sqrt3))+C`

D

`(1)/(2)tan^(-1)((e^(x)+2e^(-x))/(2))+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(f(x)-2g(x))dx`
`=int(e^(x)(e^(2x)-2))/(e^(4x)+8x^(2x)+4)dx` put `e^(x)=t`
`=int(1-(2)/(t^(2)))/((t+(2)/(t))^(2)+4)dt=(1)/(2)tan^(-1)((t+(2)/(t))/(2))+C`
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Knowledge Check

  • int(dx)/(e^(3x)+e^(-3x))=

    A
    `(1)/(3)tan^(-1)e^(-3x)+c`
    B
    `(1)/(3)tan^(-1)e^(3x)+c`
    C
    `3tan^(-1)e^(-3x)+c`
    D
    `tan^(-1)e^(3x)+c`

  • Let P(x)= int (dx)/(e^(x) + 8e^(-x) + 4e^(-3x)), Q(x)= int (dx)/(e^(3x) + 8e^(x) + 4e^(-x)) and R(x)= P(x)- 2Q (x) . If R(x)= (1)/(2) A ((B= 2e^(-x))/( C))+ K , then the value of (A, B,C) is

    A
    `(tan^(-1), 2, e^(x))`
    B
    `(tan^(-1), e^(x), 2)`
    C
    `("tan"^(-1), (1)/(2), (1)/(e^(x)))`
    D
    `("tan"^(-1), (1)/(e^(x)), (1)/(2))`

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