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The integral int (0)^(1) (dx)/(1-x+x^(2...

The integral `int _(0)^(1) (dx)/(1-x+x^(2)) ` has the value

A

`(2pi)/3`

B

`(2pi)/(sqrt(3))`

C

`(2pi)/(3sqrt(3))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(0)^(1)(dx)/(x^(2)-x+1)=int_(0)^(1)(dx)/((x-(1)/(2))^(2)+((sqrt(3))/(2))^(2))`
`=(2)/(sqrt(3))[tan^(-1)((2x-1)/(sqrt(3)))]_(0)^(1)`
`= (2)/(sqrt(3))[tan^(-1)((1)/(sqrt(3)))-tan^(-1)(-(1)/(sqrt(3)))]`
`=(2)/(sqrt(3))[(pi)/(6)-(-(pi)/(6))]=(2)/(sqrt(3)).(pi)/(3)=(2pi)/(3sqrt(3))`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  2. int (0)^(2pi) sin^(6) x cos^(5) x dx is equal to

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  3. int (-3pi//2)^(-pi//2) [ ( x + pi)^(3) + cos^(2) x ] dx is equalt to

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  4. int (0)^(3) (3x+1)/(x^(2)+9) dx =

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  5. underset(n to oo)lim underset(r=1)overset(n)sum(1)/(n)e^(r//n) is

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  6. int (-1//2)^(1//2) cos x log ((1+x)/(1-x)) dx = k log 2 , then k equ...

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  7. The value of the integral int (0)^(pi//2) sin ^(5) x dx is

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  8. If int (0)^(x^(2)) f(t) dt = x cos pix , then the value of f(4) is

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  9. If f(x) is differentiable and int0^(t^2)xf(x)dx=2/5t^5, then f(4/(25))...

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  10. The value of int (0)^(pi//2) sin ^(8) x dx is

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  11. The value of overset(pi)underset(-pi)int(1-x^(2)) sin x cos^(2) x" dx"...

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  12. int (0)^(1) (xdx)/([x + sqrt(1-x^(2))]sqrt(1-x^(2))) is equal to

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  13. The value of int(0)^(pi)(Sigma(r=0)^(3)a(r)cos^(3-r)x sin^(r)x)dx depe...

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  14. The value of int (-1)^(1) x|x| dx is

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  15. The value of integral int (1//pi)^(2//pi)(sin(1/x))/(x^(2))dx=

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  16. The value of int (1)^(e^(2)) (dx)/(x(1+ log x)^(2) ) is

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  17. int(pi//3)^(pi//2) (sqrt(1+cosx))/((1-cosx)^(5//2)) dx

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  18. The value of the integral overset(1)underset(0)int x(1-x)^(n)dx, is

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  19. The value of int (0)^(pi) (dx)/(5+4 cos x ) is

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  20. The integral int (0)^(1) (dx)/(1-x+x^(2)) has the value

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