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Evaluate the following integral: int0^1(...

Evaluate the following integral: `int_0^1(tan^(-1)x)/(1+x^2)dx`

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Given,I=`int_0^1(tan^(-1)x)/(1+x^2)dx`
Let, `tan^(−1)x=t.`
`dx=(1+x^2)dt`
Also, `x=0⇒t=tan^(−1)0=0 `and `x=1⇒t=tan^(−1)1=π/4`
​ `I=int _0^(π/4)​t dt`
`I=[t^2/2​]_0^(π/4)​`
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate the following integral: int0^(pi//4)(tan^3x)/(1+cos2x)dx

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

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

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

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

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

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

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

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

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

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  11. Evaluate the following integral: int0^(pi//2)2s in x\ cos xtan^(-1)(si...

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  12. Evaluate the following integral: int0^(pi//2)s in2xtan^(-1)(sinx)dx

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  13. int0^a xsqrt((a^2-x^2)/(a^2+x^2))dx

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  14. Evaluate the following integral: int0^(pi//2)(sin x cos x)/(cos^2x+3co...

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

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

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  17. Evaluate the following integral: int(pi//3)^(pi//2)(sqrt(1+cosx))/((1-...

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  18. Evaluate the following integral: int(-a)^asqrt((a+x)/(a-x))dx

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  19. Evaluate the following integral: int(1//3)^1((x-x^3)^(1//3))/(x^4)dx

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  20. Evaluate the following integral: int0^(pi//2)sqrt(cos x-cos^3x)"\ "(se...

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