Domanda del libro di testo
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (x^2 + 6x) / (x^2 + 3)^2 dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.6.38
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx
Guida verificata passo dopo passo1
Start by examining the expression inside the square root: \(x^2 - 4x + 5\). To simplify this, complete the square for the quadratic expression.
Rewrite \(x^2 - 4x + 5\) as \((x^2 - 4x + 4) + 1\), which simplifies to \((x - 2)^2 + 1\).
Use the substitution \(u = x - 2\), which implies $du = dx$. This changes the integral to an expression in terms of \(u\).
Rewrite the integral in terms of \(u\): replace \(x^2\) with \((u + 2)^2\) and \(\sqrt{x^2 - 4x + 5}\) with \(\sqrt{u^2 + 1}\). The integral becomes \(\int \frac{(u + 2)^2}{\sqrt{u^2 + 1}} \, du\).
Expand \((u + 2)^2\) to \(u^2 + 4u + 4\) and split the integral into three separate integrals: \(\int \frac{u^2}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{u}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{1}{\sqrt{u^2 + 1}} \, du\). Each of these can be found in standard integral tables.
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Substitution Method in Integration
The substitution method involves changing variables in an integral to simplify the integrand into a form that is easier to integrate. By choosing an appropriate substitution, often based on the inner function or expression, the integral can be transformed into a standard form found in integral tables.
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07:33
Euler's Method
Completing the Square
Completing the square is a technique used to rewrite quadratic expressions in the form ax^2 + bx + c as (x - h)^2 + k. This form simplifies the integrand, especially under square roots, making it easier to identify substitutions or match standard integral forms.
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05:22Completing the Square to Rewrite the Integrand
Using Integral Tables
Integral tables list standard integrals and their antiderivatives, providing quick references for common integral forms. After substitution and simplification, matching the integral to a form in the table allows for direct evaluation without performing integration from first principles.
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Percorso guidato
08:01Integration Using Partial Fractions
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