We can use partial fraction decomposition to solve the integral:
x^4 / [(x-1)(x^2+1)] dx
First, we factor the denominator:
x^4 / [(x-1)(x^2+1)] = A/(x-1) + Bx/(x^2+1)
Multiplying both sides by the denominator and simplifying, we get:
x^4 = A(x^2+1) + Bx(x-1)
Now, we substitute some values to find the values of A and B:
If we let x=1, then we get:
1^4 = A(1^2+1) + B(1)(1-1) = 2A
A = 1/2
If we let x=0, then we get:
0^4 = A(0^2+1) + B(0)(0-1) = A
A = 0
So, we have:
Bx(x-1) = x^4 – A(x^2+1)
Substituting the values of A and B, we get:
Bx(x-1) = x^4 – (1/2)(x^2+1)
Now, we can integrate each term separately:
∫[x^4 / (x-1)(x^2+1)] dx = ∫[1/2(x-1) + (1/2)x – (1/2)(x/(x^2+1))] dx
= 1/2 ln|x-1| + 1/4 ln(x^2+1) – 1/2 ln|x^2+1| + C
where C is the constant of integration.
