Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
√x + 1/√x

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Rewrite the given function in terms of exponents to make it easier to integrate. Recall that \(\sqrt{x} = x^{\frac{1}{2}}\) and \(\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}\). So the function becomes \(f(x) = x^{\frac{1}{2}} + x^{-\frac{1}{2}}\).

Recall the power rule for antiderivatives: for any real number \(n \neq -1\), the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

Apply the power rule to each term separately: for \(x^{\frac{1}{2}}\), add 1 to the exponent to get \(\frac{1}{2} + 1 = \frac{3}{2}\), then divide by \(\frac{3}{2}\). For \(x^{-\frac{1}{2}}\), add 1 to the exponent to get \(-\frac{1}{2} + 1 = \frac{1}{2}\), then divide by \(\frac{1}{2}\).

Write the antiderivative as the sum of the two results from the previous step, plus the constant of integration \(C\).

To check your answer, differentiate your antiderivative using the power rule for derivatives and verify that you get back the original function \(\sqrt{x} + \frac{1}{\sqrt{x}}\).

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주요 개념

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Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed with an arbitrary constant since differentiation of a constant is zero.

Power Rule for Integration

The power rule states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating functions involving powers of x, including fractional and negative exponents.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.√x + 1/√x