Question

Find the antiderivative of f(x) = 2x - (1 - x)⁴.

a) x² - x⁵/5 + C
b) x² + x⁵/5 + C
c) x² + x⁵/4 + C
d) x² - x⁵/4 + C

Answer 1

The antiderivative of f(x) = 2x - (1 - x) ^4 is obtained by integrating each term separately leading to the result x² - x⁵/5 + C.

Step-by-step explanation:

To find the antiderivative of f(x) = 2x - (1 - x)⁴, we integrate each term separately. For the first term, the power rule for integration gives us ∫x². For the second term, we'll have to expand the expression and then integrate. Expanding (1 - x) ⁴ using the binomial theorem would be quite tedious, so instead, we proceed term by term:

• The antiderivative of -1 raised to any power is just x since the derivative of a constant is 0.
• For each subsequent term of -(1 - x) ⁴, we increase the power of x by one and divide by the new power, changing the sign where necessary to reflect the negative exponent.

Integrating -(1 - x) ⁴ term by term:

1. ∫(-x) ⁴ dx = ∫ x⁴ dx = x⁵/5 (since we divide by the new power)
2. However, because we have a negative outside the term, it will be - x⁵/5.

Combining the results:

• x² from the first term
• -x⁵/5 from the second term

Therefore, the antiderivative of f(x) = 2x - (1 - x) ⁴ is x² - x⁵/5 + C.