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If y = int(0)^(x^(2))ln(1+t), then find ...

If `y = int_(0)^(x^(2))ln(1+t)`, then find `(d^(2)y)/(dx^(2))`

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The correct Answer is:
`2/(1+x^(2))[2x^(2)+(1+x^(2))ln(1+x^(2))]`
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RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -Self practive problem
  1. If f(x) = int(x)^(x^(2)) t^(2)lnt then find f'(e)

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  2. If y = int(4)^(4x^(2))t^(4)e^(4t)dt, find (d^(2)y)/(dx^(2))

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  3. If y = int(0)^(x^(2))ln(1+t), then find (d^(2)y)/(dx^(2))

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  4. If int(0)^(x^(2)(1+x))f(t)dt = x then find f(2)

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  5. Evaluate int(0)^(pi)ln(1+bcosx) dx, 'b' being parameter.

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  6. int(pi//2)^(0)sin^(11)xdx

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  7. int(0)^(-pi//2)sin^(5)xcos^(4)xdx

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  8. int(0)^(1) x^(5)sin^(-1)xdx

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  9. int(0)^(9) x(a^(2)-x^(2))^(7/2)dx

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  10. int(0)^(2) sqrt(2-x)dx.

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  11. Prove the following : int(0)^(1)e^(-x)cos^(2)xdx lt int(0)^(1)e^(-x^(2...

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  12. Prove the following : 0 lt int(0)^(pi//2)sin^(n+1)xdx lt int(0)^(pi//2...

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  13. Prove the following : e^(-(1)/(e)) lt int(0)^(1)x^(x)dx lt 1

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  14. Prove the following: -1/2lt=int0^1(x^3cosx)/(2+x^2)dx<1/2

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  15. Prove the following : 1 lt int(0)^(pi//2)sqrt(sinx)dx lt sqrt(pi/2)

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  16. Prove the following : 4/pi lt int(pi/4)^(pi/3) (tanx)/(x) lt (3sqrt(3)...

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  17. lim(nrarroo) {1/n+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+"......"+1/(...

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  18. lim(nrarroo) [1/(1+n)+(1)/(2+n)+(1)/(3+n)+"....."+(1)/(5n)]

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  19. lim(nrarroo) [sin^(3)'(pi)/(4n)+2sin^(3)'(2pi)/(4n)+3sin^(3)'(3pi)/(4n...

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  20. lim(nrarroo) sum(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))

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