Home > Calculus > Integrals of Rational Functions

What is the antiderivative of #x^3/(x^4-5)^3#?

Answer 1

# = -1/8 * 1/(x^4-5)^2 color{green}{+ C}#

You can do this without a sub, but pattern spotting, in my opinion, makes it more engaging and makes it easier to complete in your head.

first, take note of this pattern

#d/dx (1/(x^4-5)^2) = - 2 * 1/(x^4-5)^3 * 4x^3#

#= - 8 * x^3/(x^4-5)^3 #

It combines elements of the chain rule and the power rule.

and, so:

#d/dx (color{red}{-1/8} * 1/(x^4-5)^2) = x^3/(x^4-5)^3 #

additionally, the FTC has

#int dx qquad x^3/(x^4-5)^3 = -1/8 * 1/(x^4-5)^2 color{green}{+ C}#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Cameron Alexander

The antiderivative of ( \frac{x^3}{(x^4-5)^3} ) can be expressed as ( -\frac{1}{6(x^4-5)^2} + C ), where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.