Checking Antiderivative Formulas

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

∫3(2x + 1)² dx = (2x + 1)³ + C

Guida verificata passo dopo passo

Identify the integral given: \(\int 3(2x + 1)^2 \, dx\).

Recognize that the integrand is a composite function, so consider using substitution. Let \(u = 2x + 1\).

Compute the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = 2\), which implies \(dx = \frac{du}{2}\).

Rewrite the integral in terms of \(u\): \(\int 3u^2 \cdot \frac{du}{2} = \frac{3}{2} \int u^2 \, du\).

Integrate \(u^2\) with respect to \(u\): \(\int u^2 \, du = \frac{u^3}{3} + C\). Multiply by \(\frac{3}{2}\) to get \(\frac{3}{2} \cdot \frac{u^3}{3} + C = \frac{u^3}{2} + C\). Finally, substitute back \(u = 2x + 1\) to express the antiderivative in terms of \(x\).

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Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). It is represented by the indefinite integral ∫f(x) dx = F(x) + C, where C is an arbitrary constant. Understanding this helps verify if a given formula correctly reverses differentiation.

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Chain Rule and Its Reverse (Substitution Method)

The chain rule in differentiation handles composite functions. Its reverse, used in integration, often requires substitution to simplify the integral. Recognizing when to apply substitution is key to correctly finding antiderivatives of functions like (2x + 1)².

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Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. When integrating expressions like (2x + 1)², the power rule applies after appropriate substitution, ensuring the integral is computed correctly rather than just raising the inner function to a higher power.

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