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Show that int(0)^(a)x(a-x)^(n)dx=(a^(n+2...

Show that `int_(0)^(a)x(a-x)^(n)dx=(a^(n+2))/((n+1)(n+2))`.
Hence find `int_(0)^(2) x sqrt(2- x )dx`

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Verified by Experts

The correct Answer is:
`(16sqrt(2))/(15)`
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AAKASH SERIES-DEFINITE INTEGRALS-Exercise-2.3 (Level-1)
  1. int((pi)/(5))^((3pi)/(10))(f(sin x))/(f(sin x) + f(cos x)) dx

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  2. Find underset(-a)overset(a)int(x^(2)+sqrt(a^(2)-x^(2)))dx

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  3. Evaluate underset(-pi//2)overset(pi/2)int sin|x| dx

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  4. Evaluate the following integrals (iii) int(-1)^(1)|x|dx

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  5. int(pi)^((pi)/(2)) sin|2x|dx

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  6. Evaluate the following integrals (iv) int(-1)^(1)(x^(4)+sin^(3)x)dx

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  7. Evaluate the definite integrals . underset(-pi//2)overset(pi//2)int ...

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  8. int(0)^(2pi)(1)/(1+e^(sinx))dx=

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  9. int((-pi)/(4))^((pi)/(4)) (dx)/(1+e^(tan x))

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  10. int(-1)^(1) (cos h x)/(1+e^(2x))dx=

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  11. Show that int(0)^(a)x(a-x)^(n)dx=(a^(n+2))/((n+1)(n+2)). Hence find ...

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  12. Evaluate the integral underset(0)overset(pi//4)int log(1+tanx)dx

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  13. Evaluate the integral underset(0)overset(1)int (log(1+x))/(1+x^(2))d...

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  14. Evaluate (i) int(0)^(pi/2) (sin^(2)x)/(sin x + cos x ) dx

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  15. int(0)(pi//2) ( x dx)/(Sin x + Cos x)=

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  16. Evaluate (i) int(0)^(pi) ( x sin x)/(1+ sin x) dx

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  17. Evaluate the integral underset(0)overset(pi)int (xsinx)/(1+cos^(2)x)...

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  18. Evaluate (iii) int(0)^(pi)(x sin^(3)x)/(1+cos^(2)x)dx

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  19. Evaluate the integral underset(0)overset(pi)int x sin^(3)x dx

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  20. Evaluate int(0)^(pi) sin^(3) theta(1+2 cos theta)(1+ cos theta)^(2) d ...

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