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Let a , b , c be nonzero real numbers su...

Let `a , b , c` be nonzero real numbers such that `int_0^1(1+cos^8x)(a x^2+b x+c)dx` `=int_0^2(1+cos^8x)(a x^2+b x+c)dx=0` Then show that the equation `a x^2+b x+c=0` will have one root between 0 and 1 and other root between 1 and 2.

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CENGAGE PUBLICATION-INTEGRALS-All Questions
  1. The value of the definite integral int(0)^(1)(1+e^(-x^(2))) dx is

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  2. Prove that 0 lt int0^1(x^7dx)/((1+x^8)^(1/3)) lt 1/8

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  3. Let a , b , c be nonzero real numbers such that int0^1(1+cos^8x)(a x^2...

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  4. Evaluate: 5050(int0^1(1-x^(50))^(100)dx)/(int0^1(1-x^(50))^(101)dx)

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  5. Ifint(pi/3)^xsqrt((3-sin^2t))dt+int0^ycostdt=0,then evaluate (dy)/(d...

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  6. If f(x)=int0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s (a)f(x)+f...

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  7. Evaluate: int(-pi/2)^(pi/2)log((a-sin theta)/(a+sin theta)) d theta,a ...

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  8. Evaluate: int(0)^1(dx)/(1+x^2)

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  9. If f(t) is an odd function, then prove that varphi(x)=inta^xf(t)dt is ...

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  10. Evaluate: int2^3(2x^5+x^4-2x^3+2x^2+1)/((x^2+1)(x^4-1))dx

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  11. Find the value of int(0)^(1){(sin^(-1)x)//x}dx.

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  12. Show that int0^(npi+v)|sinx|dx=2n+1-cosv , where n is a positive int...

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  13. Find I=int0^pi ln(1+cosx)dx

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  14. The value of inta^b(x-a)^3(b-x)^4dx is

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  15. The value of the integral int(-1//2)^(1//2) {((x+1)/(x-1))^(2)+((x-1...

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  16. The value of int(-a)^a(cos^(- 1)x-sin^(- 1)sqrt(1-x^2))dx is (a>0) w...

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  17. If f is an odd function, then evaluate I=int(-a)^a(f(sinx)dx)/(f(cosx...

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  18. Evaluate : int0^(pi/4)log(1+tanx)dx.

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  19. Evaluate: int(-pi/4)^(pi/4)(x^9-3x^5+7x^3-x+1)/(cos^2x)dx

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  20. Let a+b=4,w h e r ea<2,a n dl e tg(x) be a differentiable function. If...

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