Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫xsinx dx = -x cos x + C

Verified step by step guidance

1. Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
2. Identify parts for integration by parts: let \(u = x\) (so \(du = dx\)) and \(dv = \sin x \, dx\) (so \(v = -\cos x\)).
3. Apply the formula: \(\int x \sin x \, dx = -x \cos x - \int (-\cos x) \, dx = -x \cos x + \int \cos x \, dx\).
4. Integrate \(\int \cos x \, dx\) to get \(\sin x\), so the full antiderivative is \(-x \cos x + \sin x + C\).
5. Compare this with the given formula \(\int x \sin x \, dx = -x \cos x + C\); since the \(\sin x\) term is missing, the given formula is incorrect.

Key Concepts

Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). The indefinite integral symbol ∫ represents the family of all antiderivatives, including a constant of integration C, since differentiation of a constant is zero.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du, where u and dv are parts of the original integrand chosen to simplify the integral.

Verification of Antiderivatives

To verify if a given formula is an antiderivative, differentiate the proposed function and check if the result matches the original integrand. This method confirms correctness by reversing the integration process.