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Evaluate the following integral: int0^(p...

Evaluate the following integral: `int_0^(pi//4)sin^3 2tcos2t\ dt`

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Let,I=`int_0^(pi//4)sin^3 2tcos2t\ dt`
let `sin2t=u. `
Then, `cos2tdt=1/2​du`
Also, `t=0⇒u=sin0=0` and` t=π/4​⇒u=sin(π/2)​=1`
`I=1/2int_0^1 ​u^3du`
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate the following integral: int4^(12)x(x-4)^(1//3)dx

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

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

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

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

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

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

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

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

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

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

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

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

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

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  15. Evaluate the following integral: int0^(pi//2)x^2sinx\ dx

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

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  17. Evaluate the following integral: int0^(pi//2)(cos^2x)/(1+3sin^2x)dx

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  18. Evaluate the following integral: int0^pi5(5-4costheta)^(1//4)sinthetad...

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  19. Evaluate the following integral: int4^9(sqrt(x))/((30-x^(3/2))^2)dx

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  20. Evaluate the following integral: int0^pisin^3x(1+2cos)(1+cos x)^2dx

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