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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Identify a substitution that simplifies the integral. Notice that the denominator is \( (x^2 + 3)^2 \) and the numerator contains terms \( x^2 + 6x \). Consider letting \( u = x^2 + 3 \) because its derivative relates to the numerator.

Compute the derivative of \( u \) with respect to \( x \): \( \frac{du}{dx} = 2x \), which implies \( du = 2x \, dx \). This will help express parts of the integral in terms of \( u \) and \( du \).

Rewrite the integral in terms of \( u \) and \( du \). Express \( x^2 + 6x \) in terms of \( u \) and \( x \), and replace \( dx \) using \( du = 2x \, dx \) or \( dx = \frac{du}{2x} \). This step may require splitting the integral or manipulating the numerator to match the substitution.

After substitution, simplify the integral to a form involving \( u \) only, ideally matching a standard integral from the table, such as \( \int \frac{du}{u^2} \) or similar.

Integrate the simplified expression with respect to \( u \), then substitute back \( u = x^2 + 3 \) to express the answer in terms of \( x \).

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주요 개념

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Integration by Substitution

Integration by substitution involves changing variables to simplify an integral into a more recognizable form. By letting a new variable equal a function inside the integral, the integral can often be rewritten in terms of this variable, making it easier to evaluate using standard formulas or tables.

Recognizing Derivatives within the Integral

Identifying parts of the integrand that correspond to the derivative of another part is crucial for substitution. For example, noticing that the numerator resembles the derivative of the denominator helps in choosing an appropriate substitution, simplifying the integral into a standard form.

Using Integral Tables

Integral tables provide formulas for common integrals, allowing quick evaluation once the integral is transformed into a standard form. After substitution simplifies the integral, matching it to a table entry enables direct application of known results without performing integration from scratch.

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