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If inta^b x^3dx=0, and If inta^b x^2d...

If `int_a^b x^3dx=0`, and If `int_a^b x^2dx=2/3,` find a and b.

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Given, `int_a^b x^3dx=0`
=`[x^4/4]_a^b=(b^4-a^4)/4`
=`b^4=a^4`
=`b=+-a`
Also, `int_a^b x^2dx=2/3`
`[x^3/3]_a^b=2/3`
=`b^3-a^3=2`
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. If int0^1(3x^2+2x+k)dx=0, find the value of kdot

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  2. If int1^a (3x^2 + 2x +1)dx = 11 then the value of a is

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  3. If inta^b x^3dx=0, and If inta^b x^2dx=2/3, find a and b.

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  4. Evaluate: int0^(pi//2)sqrt(1-cos2x)dxdot

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  5. Evaluate: int0^(pi//2)sin^4xdx

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  6. Evaluate: int0^(pi//4)sqrt(1+sin2x)dx (ii) int0^(pi//4)sqrt(1-sin2x...

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  7. Evaluate: int(pi//4)^(pi//2)sqrt(1-sin2x)dx

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  8. Evaluate the following integrals: (1-35) int(-2)^2(3x^3+2|x|+1)/(x^...

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  9. Evaluate the following integrals: (1-35) int0^pixsinx cos^2x dx

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  10. Show that: int0^(pi//2)f(sin2x)sinxdx=sqrt(2)int0^(pi//4)f(cos2x)co...

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  11. For x >0,l e tf(x)=int1^x((log)e t)/(1+t)dtdot Find the function...

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  12. Evaluate the following integrals: (1-35) int0^(pi//2)(sin^n x)/(sin...

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  13. Prove that: int0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx= pi^2

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  14. Prove that: int0^(pi//2)log|tanx+cotx|dx=pi(log)e2

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  15. Evaluate: int(-2)^2|xcospix|dx

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  16. Evaluate: int(pi//4)^(pi//4)log(sinx+cosx)dx

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  17. Evaluate: int0^pixlogsinxdx

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  18. If int0^pi 1/(a+bcosx)dx=pi/sqrt(a^2-b^2), then int0^pi 1/(a+bcosx)^2d...

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  19. int0^1sqrt((1-x)/(1+x))dx= (pi)/2 (b) (pi)/2-1 (c) (pi)/2+1 (d) ...

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  20. int0^(pi//2)(cosx)/((2+sinx)(1+sinx))dx equals (a)log(2/3) (b) l...

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