24most probably its not in syllabus...
Here β represents beta function
Q1 If \int_{0}^{n}{(1-\frac{x}{n})^n}x^{k-1}dx=R.\beta (k,n+1) find R
Q2 If \int_{0}^{\infty}{\frac{x^{m-1}}{(1+x)^{m+n}}}dx=k.\int_{0}^{\infty}{\frac{x^{n-1}}{(1+x)^{m+n}}}dx ,find value of k
Q3 The value of \int_{0}^{1}x^{m-1}(1-x^p)^{n-1}dx is equal to ??
a)p . β(m/p,n)
b)1/p . β(mp,n)
c)1/p . β(m,np)
d)1/p . β(m/p,n)
Q4 \int_{0}^{2}(8-x^3)^{-1/3}dx=k.\beta(1/3,2/3) ,find value of k
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8 Answers
1why are you giving questions not in syllabus???focus on what's in syllabus & u will reap good results.
i did all these but even then i secured a rank of 4123 in jee-09.
HOWEVER I AM SOLVING THEM.
let X/N =T THEN ,dX=NdT
hence(0 to 1) ∫((1-T)^n)T^k-1 (n^k-1)*n dT
=n^K (0 to 1)∫((1-T)^n)t^k-1 dT
=n^K*β(N+1,K)
HENCE R=n^K.
1\int_{0}^{1}{x^{m-1}}(1-x^{p)}^{n-1}dx
\int_{0}^{1}{x^{m-1}}(1-x^{p)}^{n-1}dx let x^{p}=t hence, px^{p-1}dx=dt hence\frac{1}{p}\int_{0}^{1}{t^{\frac{m}{p}-1}(1-t)^{n-1}dt} =\frac{1}{p}\beta (\frac{m}{p},n)
24hey dude,,,maybe u didnt read post#2..and the topic of this post[3]
i know these ques are not in syll...posted them to know whether tehy should be done on a safer side or not...
1But, these ques can come in comprehension type ques.....!
Ans (2)> k = 1 cos the LHS represents β(m, n) & RHS represents k. β(n, m).
Since, β(m, n) = β(n, m) => k = 1.
Here, put x = t/(t+1).
Hint:-
=> dx = dt/(t+1)2.
Now, just solve it u vl get LHS = β(m, n) & RHS = k. β(n, m).
1For the IVth problem, take 8 out of the bracket & then put x/2 = t.
integration limit reduces to 0 to 1.
Now, u can easily apply Beta function here!