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Evaluate the following integral: int0^pi...

Evaluate the following integral: `int_0^pi1/(5+3cosx)dx`

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Verified by Experts

Given`int_0^pi1/(5+3cosx)dx`
=`int_0^pi 1/(5+(3−3tan^2(x/2)​)​/(1-tan^2(x/2​))`
​=`int_0^pi (1+tan^2(x/2)​​)/(5+5tan^2(x/2)​+3−3tan^2(x/2))`
​ ​=`int_0^pi (sec^2(x/2)​​)/(8+2tan^2(x/2)​)`
​ Take `tanx​/2=t⟹1/2​.sec^2(x/2)​dx=dt`
Then, we get
`∫dt/(4+t^2)`
`=1/2​[tan^(−1)(t/2)]`
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate the following integral: int0^1tan^(-1)x\ dx

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  2. Evaluate the following integral: int0^(pi//4)(sqrt(t a n x)+sqrt(cotx)...

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  3. Evaluate the following integral: int0^pi1/(5+3cosx)dx

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  4. Evaluate the following integral: int0^(pi//2)(x+sinx)/(1+cosx)dx

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  5. Evaluate the following integral: int0^(pi//4)(sin x+cosx)/(3+sin2x)dx

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  6. Evaluate the following integral: int0^1(1+x^2)/(x^4+x^2+1)dx

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  7. Evaluate the following integral: int4^(12)x(x-4)^(1//3)dx

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  8. Evaluate the following integral: int(-1)^1 5x^4sqrt(x^5+1)dx\

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  9. Evaluate the following integral: int0^(pi//4)sin^3 2tcos2t\ dt

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  10. Evaluate the following integral: int0^(pi//2)(cos x)/(1+sin^2x)dx

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  11. Evaluate the following integral: int0^(pi//3)(cos\ x)/(3+4sinx)dx

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  12. Evaluate the following integral: int0^2x\ sqrt(x+2)\ dx

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  13. Evaluate the following integral: int0^(pi//2)(sinxcosx)/(1+sin^4x)dx

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  14. Evaluate the following integral: int0^pi1/(5+4sinx)dx

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  15. Evaluate the following integral: int0^pi1/(3+2sinx+cosx)dx

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  16. Evaluate the following integral: int0^(1//2)(xsin^(-1)x)/(sqrt(1-x^2))...

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  17. Evaluate the following integral: int0^(pi//4)(tan^3x)/(1+cos2x)dx

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  18. Evaluate the following integral: int0^(pi)x/(a^2cos^2x+b^2sin^2x)dx

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  19. Evaluate the following integral: int0^1(tan^(-1)x)/(1+x^2)dx

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  20. Evaluate the following integral: int0^1x t a n^(-1)x\ dx

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