Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ √x e√x dx

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Start by identifying a substitution that simplifies the integral. Notice the presence of both \( \sqrt{x} \) and \( e^{\sqrt{x}} \). Let’s set \( t = \sqrt{x} \), which means \( t = x^{1/2} \).

Express \( x \) and \( dx \) in terms of \( t \). Since \( t = x^{1/2} \), then \( x = t^2 \). Differentiate both sides with respect to \( t \) to find \( dx \): \( dx = 2t \, dt \).

Rewrite the integral \( \int \sqrt{x} e^{\sqrt{x}} \, dx \) in terms of \( t \): \( \sqrt{x} = t \), \( e^{\sqrt{x}} = e^{t} \), and \( dx = 2t \, dt \). So the integral becomes \( \int t \cdot e^{t} \cdot 2t \, dt = \int 2t^{2} e^{t} \, dt \).

Now, focus on evaluating \( \int 2t^{2} e^{t} \, dt \). You can factor out the constant 2: \( 2 \int t^{2} e^{t} \, dt \). This integral requires integration by parts, where you can let \( u = t^{2} \) and \( dv = e^{t} dt \).

Apply integration by parts: compute \( du = 2t \, dt \) and \( v = e^{t} \). Then use the formula \( \int u \, dv = uv - \int v \, du \) to break down the integral further. You may need to apply integration by parts a second time to fully evaluate the integral.

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function, such as e^(√x).

Properties of Exponential Functions

Understanding the properties of exponential functions, like e^(√x), is crucial for integration. These functions often require special techniques because their derivatives involve the chain rule. Recognizing how to differentiate and integrate exponential functions helps in manipulating the integral effectively.

Algebraic Manipulation of Radicals

Algebraic manipulation of radicals involves rewriting expressions with roots, such as √x, in a form that is easier to integrate, often as x raised to a fractional power. This simplification allows the use of standard integration rules and makes substitution or integration by parts more straightforward.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ √x e√x dx