Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.2.46

Capitolo 8, Problema 8.2.46

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

Guida verificata passo dopo passo

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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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).

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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.

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Properties of Functions

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.

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