Markiyan Hirnyk

Markiyan Hirnyk
9 years, 227 days


These are questions asked by Markiyan Hirnyk

How to calculate the integral

(symbolically or/and numerically) with Maple?

 

Let us consider the definite integral

J:=int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1);

Maple fails with it, Mathematica 10.1 finds it in terms of  special functions. Let us look at the integrand:
plot(x-(-x^5+1)^(1/5), x = 0 .. 1);

We see the expression under the modulus changes its sign at the unique point of RealRange(0,1). Therefore

solve(x-(-x^5+1)^(1/5));


Then

J:= int(-x+(-x^5+1)^(1/5), x = 0 .. (1/2)*2^(4/5))+int(x-(-x^5+1)^(1/5), x = (1/2)*2^(4/5) .. 1);

which outputs a complicated expression

(1/8)*2^(4/5)*(4*hypergeom([-1/5, 1/5], [6/5], 1/2)-2^(4/5))+(1/2)*2^(4/5)*((1/2)*2^(1/5)-(1/4)*2^(4/5))-(1/25)*Pi*csc((1/5)*Pi)*(-(25/2)*sin((1/5)*Pi)*GAMMA(4/5)*2^(4/5)*hypergeom([-1/5, 1/5], [6/5], 1/2)/Pi+(5/4)*sec((3/10)*Pi)*cos((1/10)*Pi)*2^(3/5)*Pi^(1/2)*csc((3/10)*Pi)/GAMMA(7/10))/GAMMA(4/5).

At the same time we have

int(abs(x-(-x^5+1)^(1/5)), x = 0 .. 1, numeric);

                          0.5000000000

How to obtain 1/2 symbolically?






How to express sin(-(1/6)*Pi+(1/2)*arccos(1/3)) in radicals with Maple? The use of the applyrule command is not desired. Here is one of  my tries:
>convert(expand(sin(-(1/6)*Pi+(1/2)*arccos(1/3))), radical);
-(1/2)*cos((1/2)*arccos(1/3))+(1/2)*sqrt(3)*sin((1/2)*arccos(1/3))

of a square matrix having its entries 1,2,...16? See http://oeis.org/A085000 for info.

How to find the maximum natural number n s. t. the sum of its cubed digits is greater than or equal to n? Of course, with Maple. The same question for the sum of the  digits to k-th power. Here are my unsuccessful attempts:
1.Optimization:-Maximize(n, {n <= convert(map(c ->c^3, convert(n, base, 10)), `+`)}, assume = integer);

Error, invalid input: `convert/base` expects its 1st argument, n, to be of type {integer, list(integer)}, but received n


2. for n while n <= convert(map(c ->c^3, convert(n, base, 10)), `+`) do print(n) end do;

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