Comparing Antiderivatives (a) Explain why the antiderivative...

Comparing Antiderivatives
(a) Explain why the antiderivative $y_{1}=e^{x+C_{1}}$ is equivalent to the antiderivative $y_{2}=C e^{x} .$
(b) Explain why the antiderivative $y_{1}=\sec ^{2} x+C_{1}$ is equivalent to the antiderivative $y_{2}=\tan ^{2} x+C .$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Constant of Integration

When finding antiderivatives, an arbitrary constant is added to account for the fact that differentiating a constant gives zero. This means that any two antiderivatives that differ only by a constant represent the same family of functions, making them equivalent representations of the indefinite integral.

Properties of Exponents

Exponential functions satisfy the rule e^(a+b) = e^a · e^b. This property allows us to factor out a constant from the exponent as a multiplicative constant, which then becomes absorbed into the arbitrary constant of integration, showing the equivalence of expressions like e^(x+C1) and C e^x.

Trigonometric Identities

Fundamental trigonometric identities, such as sec^2 x = tan^2 x + 1, are used to manipulate antiderivative expressions. By recognizing that a constant term (like 1) can be absorbed into the arbitrary constant, different looking antiderivatives, like sec^2 x + C1 and tan^2 x + C, can be shown to represent the same family of functions.

Transcript

Please give Ace some feedback