Evaluate the integral: ∫01(1−x73−1−x37)dx

Question

Evaluate the integral:  ∫01(1−x73−1−x37)dx

encolatgehu · Accepted Answer

Let m,n&amp;gt;0. Then observe that  ∫011−xmndx  is the area of the region given by inequalities  0≤x≤1 and 0≤y≤1−xmn  But the last inequality is equivalent to 0≤xm+yn≤1. Thus  ∫011−xmndx=[Area given by  0≤xm+yn≤1, 0≤x, y≤1]  Thus by interchanging the role of x and y, we have  ∫011−xmndx=∫011−xnmdx  Of course, we can give a purely analytic approach. Let y=1−x73. Then x=1−y37 and hence by integration by substitution,  ∫011−x73dx=∫01y(x)dx  =∫10ydx(y)  =[yx(y)]10−∫10x(y)dy  =∫011−y37dy

Serita Dewitt · Accepted Answer

Another way to go is to use β-function  β(x,y)=∫01tx−1(1−t)y−1dt, Re(x), Re(y)&amp;gt;0  ∫011−x73dx=17∫01u−67(1−u)13dx=…  Now, you can finish the problem.

nick1337 · Accepted Answer

Observing the inverse of the function f(x)=1−x73 on the interval [0,1] is f(x)=1−x73, using the result ∫abf(x)dx+∫f(a)f(b)f−1(x)dx=bf(b)−af(a) since f(a)=f(0)=1 and f(b)=f(1)=0 it immediately follows that ∫011−x73dx+∫101−x37dx=0 or ∫01[1−x73−1−x37]dx=0

Evaluate the integral: \int_0^1(\root(3)(1-x^7)-\root(7)(1-x^3))dx

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