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Evaluate the following integrals: int5...

Evaluate the following integrals:
`int5^(5^(5^(x)))*5^(5^(x))*5^(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
`(5^(5^(5^(x))))/((log5)^(3))+C`
Putting `5^(x)=thArr5^(x)log5dx=dthArr5^(x)dx=(1)/(log5)dt`.
`:.I=int5^(5^t)5^(t)(1)/(log5)dt=int5^(u)*(1)/((log5)^(2))"du, where "5^(t)=uhArr5^(t)log5dt=du`
`=int(1)/((log5)^(3))"du, where "5^(u)=vhArr5^(u)log5du=dv`
`(v)/((log5)^(3))+C(5^(u))/((log5)^(3))+C=(5^(5^(t)))/((log5)^(3))+C=(5^(5^(5^(x))))/((log5)^(3))+C`.
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RS AGGARWAL-METHODS OF INTEGRATION -Exercise 13C
  1. Evaluate: inte^x\ ((sin4x-4)/(1-cos4x))\ dx

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  2. Evaluate the following integrals: int(e^(x)[sqrt(1-x^(2))sin^(-1)x+1...

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  3. Evaluate the following integrals: inte^(x)((1+xlogx)/(x))dx

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  4. Evaluate the following integrals: inte^(x)*(x)/((1+x)^(2))dx

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  5. Evaluate the following integrals: inte^(x)((x-1))/((x+1)^(3))dx

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  6. Evaluate the following integrals: inte^(x)((2-x))/((1-x)^(2))dx

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  7. Evaluate the following integrals: inte^(x)*((x-3))/((x-1)^(3))dx

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  8. Evaluate the following integrals: inte^(3x)((3x-1)/(9x^(2)))dx

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  9. Evaluate the following integrals: int((x+1))/((x+2)^(2))e^(x)dx

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  10. int(x e^(2x))/((1+2x)^2)dx

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  11. Evaluate the following integrals: inte^(2x)((2x-1)/(4x^(2)))dx

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  12. Evaluate the following integrals: inte^(x)(logx+(1)/(x^(2)))dx

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  13. Evaluate the following integrals: int(logx)/((1+logx)^(2))dx

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  14. int[sin(logx)+cos(logx)]dx

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  15. Evaluate the following integrals: int{(1)/(logx)-(1)/((log)^(2))}dx

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  16. Evaluate the following integrals: int{log(logx)+(1)/((logx)^(2))}dx

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  17. Evaluate: int(sin^(-1)sqrt(x)-cos^(-1)sqrt(x))/(sin^(-1)sqrt(x)+cos^(-...

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  18. Evaluate the following integrals: int5^(5^(5^(x)))*5^(5^(x))*5^(x)dx

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  19. Evaluate the following integrals: inte^(2x)((1+sin2x)/(1+cos2x))dx

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  20. Evaluate the following integrals: inte^(2x)((1+sin2x)/(1+cos2x))dx

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