Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

Verified step by step guidance

1. Identify the given antiderivative formula: \(\int \csc^{2}\left(\frac{x - 1}{3}\right) \, dx = -3 \cot\left(\frac{x - 1}{3}\right) + C\).
2. Recall that to verify an antiderivative, you differentiate the right-hand side and check if you get the original integrand.
3. Differentiate the function \(-3 \cot\left(\frac{x - 1}{3}\right)\) using the chain rule. Start by differentiating \(\cot(u)\) where \(u = \frac{x - 1}{3}\).
4. Use the derivative formula \(\frac{d}{du} \cot(u) = -\csc^{2}(u)\), so \(\frac{d}{dx} \cot\left(\frac{x - 1}{3}\right) = -\csc^{2}\left(\frac{x - 1}{3}\right) \cdot \frac{d}{dx} \left(\frac{x - 1}{3}\right)\).
5. Calculate \(\frac{d}{dx} \left(\frac{x - 1}{3}\right) = \frac{1}{3}\), then multiply all parts together and simplify to confirm the derivative equals \(\csc^{2}\left(\frac{x - 1}{3}\right)\), which verifies the formula.

Key Concepts

Antiderivatives and Indefinite Integrals

An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C, since differentiation of a constant is zero.

Differentiation of Composite Functions (Chain Rule)

The chain rule is used to differentiate composite functions, where one function is inside another. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). This is essential when verifying antiderivatives involving expressions like (x - 1)/3.

Derivatives of Trigonometric Functions

Knowing the derivatives of basic trig functions like cotangent and cosecant squared is crucial. For example, the derivative of cot(u) is -csc²(u) times the derivative of u. This knowledge helps verify if the given antiderivative formula is correct by differentiation.