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If P = int (0) ^(oo) (x ^(2))/(1+ x ^(4)...

If `P = int _(0) ^(oo) (x ^(2))/(1+ x ^(4)) dx , Q = int _(0) ^(oo) (x dx )/(1 + x ^(4)) and R = int _(0) ^(oo) (dx )/(1+ x ^(4))` then prove that
(a) `Q = pi/4`
(b) ` P =R`
(c ) `P- sqrt2 Q + R = (-pi)/(2 sqrt2)`

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The correct Answer is:
` int _(0) ^(1)
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MOTION-DEFINITE INTEGRATION -EXERCISE -3
  1. If P = int (0) ^(oo) (x ^(2))/(1+ x ^(4)) dx , Q = int (0) ^(oo) (x dx...

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  2. int (0) ^(1) (sin^-1(x))/( sqrt(1 - x ^(2)))dx

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  3. int (0) ^(2pi) ( sin 2x )dx

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  4. Evaluate : int0^(2pi)e^xcos(pi/4+x/2)dx

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  5. Evaluate: int (0) ^(1) e ^(ln tan ^(-1)) x. sin ^(-1) (cos x ) dx.

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  6. The range of the function f(x)=int(-1)^(1)(sinxdt)/(1+2tcosx+t^(2)) is

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  7. Evaluate I (n) = int (1) ^(e) (ln ^(n) x) dx hence find I (3).

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  8. int0^sqrt3 sin^-1\ (2x)/(1+x^2)\ dx

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  9. For a >= 2, if the value of the definite int0^oo(dx)/(a^2+(x-(1/ x))^2...

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  10. The value of underset(0)overset(1)int({2x}-1)({3x}-1)dx, (where {} de...

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  11. Find the value of the definite integral int0^pi | sqrt (2)sin x+2 cos ...

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  12. Evaluate the integral int (3) ^(5) (sqrt(x +2 sqrt(2x-4))+ sqrt(x-2 ...

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  13. Find the value of int(-pi/3)^(pi/3)(pi+4x^3)/(2-cos(|x|pi/3))dx

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  14. Evaluate : int- 1^1((2x^(332)+x^(998)+4x^(1668)*sinx^(691))/(1+x^(666)...

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  15. int (-1) ^(2) (x ^(2) -x)/(sqrt(x ^(2) + 4))

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  16. int (0) ^(pi//4) (xdx )/(cos x (cos x + sin x ))

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  17. Evaluate :int0^(pi/2)(asinx+bcosx)/(sin(pi/4+x))dx

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  18. int (0) ^(pi) ((ax +b) sec c tan x)/(4 + tan ^(2) x ) dx (a,b gt 0)

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  19. int0^pi((2x+3)sinx)/((1+cos^2x)dx

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  20. intxsin^(- 1)[1/2sqrt((2a-x)/a)]dx

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