Question #523

What is the antiderivative for x^2(x^2-1)^1/2?

Expert's answer

Expression x^m(bx^n+a)^p (1) is called binomial differential. We can convert your expression into following: x^2(bx^2-1)^1/2; in your particular case m=2, n=2, p=1/2. As it was proved by Chebyshev binomial differential has antiderivative expressed by elementary functions only in the following 3 cases:

1) p-integer, (m+1)/n - a fractional, rational;

2) p-fractional, rational, (m+1)/n - integer;

3) p-fractional, rational, (m+1)/n - fractional, rational, but p+((m+1)/n) - integer.

In other cases, the integral can not be calculated in terms of elementary functions.

Your case is #3. Use next substitution: a+b(x^n)=(x^n)(u^2). Making the change of variable, solve and return to the old variable.

1) p-integer, (m+1)/n - a fractional, rational;

2) p-fractional, rational, (m+1)/n - integer;

3) p-fractional, rational, (m+1)/n - fractional, rational, but p+((m+1)/n) - integer.

In other cases, the integral can not be calculated in terms of elementary functions.

Your case is #3. Use next substitution: a+b(x^n)=(x^n)(u^2). Making the change of variable, solve and return to the old variable.

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