Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx

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Identify the integral to be evaluated: \(\int_0^3 (\sqrt{2} + 1) x^{\sqrt{2}} \, dx\).

Since \((\sqrt{2} + 1)\) is a constant coefficient, factor it out of the integral: \( (\sqrt{2} + 1) \int_0^3 x^{\sqrt{2}} \, dx\).

Recall the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).

Apply the power rule to \(\int_0^3 x^{\sqrt{2}} \, dx\), which becomes \(\left[ \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} \right]_0^3\).

Multiply the result by the constant \((\sqrt{2} + 1)\) and then evaluate the expression at the upper limit \(x=3\) and lower limit \(x=0\), subtracting the two values to find the definite integral.

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주요 개념

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Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Power Rule for Integration

The power rule states that ∫ x^n dx = (x^(n+1)) / (n+1) + C for any real number n ≠ -1. This rule is essential for integrating functions where the variable is raised to a constant power, including irrational exponents like √2.

Constant Multiples in Integration

When integrating, constants can be factored out of the integral to simplify calculations. For example, ∫ c·f(x) dx = c ∫ f(x) dx, where c is a constant. This allows the integral of (√2 + 1) x^(√2) to be treated as (√2 + 1) times the integral of x^(√2).

Evaluate the integrals in Exercises 97–110.99. ∫₀³ (√2 + 1)x^(√2) dx