I came across this problem in my homework, and I was wondering if partial fractions would be rational for this problem.
Int [(2x)/((x^2)^2)]dx
If I don't use partial fractions, what would I use?
INT 2x/x^4 dx = INT 2/x^3 dx=-1/2 x^-4 + C
I messed up. I meant (x^2+1)^12
To solve the integral ∫[2x/((x^2)^2)] dx, you can use a u-substitution. However, partial fractions would not be applicable in this case because the denominator is a power of a polynomial, not a polynomial itself.
Instead, you can simplify the expression by simplifying the denominator. Notice that (x^2)^2 is equivalent to x^4. Therefore, the integral can be rewritten as ∫[2x/x^4] dx.
Next, you simplify the expression by canceling out one of the x's in the numerator and denominator. This leaves you with ∫[2/x^3] dx.
To integrate this expression, you can now use the power rule for integration. The power rule states that the integral of x^n with respect to x is equal to (1/(n+1)) * x^(n+1), where n is any real number except -1.
Applying the power rule to ∫[2/x^3] dx, you increase the exponent by 1 and divide by the new exponent:
∫[2/x^3] dx = (2/(-3+1)) * x^(-3+1) = (-1/2) * x^-2 = (-1/2) * (1/x^2) + C,
where C is the constant of integration.
Therefore, the integral of [(2x)/((x^2)^2)] dx is (-1/2) * (1/x^2) + C.