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Evaluate int(0)^(pi)xsin^(7)xcos^(6)xdx...

Evaluate `int_(0)^(pi)xsin^(7)xcos^(6)xdx`

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Verified by Experts

Let `I = underset(0)overset(pi)intx sin^(7)x cos^(6)x dx`
`I = underset(0)overset(pi)intsin^(7)(pi-x)cos^(6)(pi-x)dx = piunderset(0)overset(pi)intsin^(7)xcos^(6)xdx - underset(0)overset(pi)intxsin^(7)x cos^(6) x dx`
`rArr 2I = pi.2 underset(0)overset(pi/2)intsin^(7)x.cos^(2)xdx rArr I = pi(6.4.2.5.3.1)/(13.11.9.7.5.3.1) = I = (48pi)/(9009)`
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RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -High Level Problem
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  2. Find the integral value of a for which int(0)^(pi//2)(sinx+acosx)^(3) ...

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  3. Evaluate int(0)^(pi)sqrt((cosx+cos2x+cos3x)^(2)+(sinx+sin2x+sin3x)^(...

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  4. Let alpha & beta be distinct positive roots of the equation tanx = 2x,...

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  5. Evaluate : lim(ararr(pi/2)^(-)) int(0)^(a)(cosx)ln(cosx)dx

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  6. Find the value of a(0 lt a lt 1) for which the following definite inte...

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  9. Evaluate Lim(n->oo)n^2 int(-1/n)^(1/n) (2006sinx+2007cosx) |x|dx

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  10. Let sequence (a(n)) be defined as a(1) = pi/4, a(n) = int(0)^(1//...

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  11. Find f(x) if it satisfies the relation f(x) = e^(x) + int(0)^(1) (x...

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  12. Evaluate : int(-1//sqrt(3))^(1//sqrt(3))(x^(4))/(1-x^(4)) cos^(-1) ((2...

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  13. Evaluate int(0)^(1)(1)/((5+2x-2x^(2))(1+e^((2-4x)))dx

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  14. Prove that for any positive integer k , (sin2kx)/(sinx) = 2 [cosx+co...

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  15. If n gt 1. Evaluate int(0)^(oo)(dx)/((x+sqrt(1+x^(2)))^(n))

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  16. Let f(x) be a continuous function AAx in R , except at x=0, such that...

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  17. Given that lim(nrarroo) sum(r=1)^(n) (log(e)(n^(2)+r^(2))-2log(e)n)/n ...

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  18. For a natural the following questions : Now answer the following que...

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  19. Given that lim(x to oo)(int(0)^(x)(t^(2))/(sqrt(a+t))dt)/(bx-sinx) = 1...

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  20. Draw a graph of the function f(x) = cos^(-1) (4x^(3)-3x), x in [-1,1] ...

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  21. Consider a square with vertices at (1,1),(-1,1),(-1,-1),a n d(1,-1)dot...

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