Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx

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Start by recognizing that the integral is of the form \(\int x (7x + 5)^{3/2} \, dx\), which suggests using substitution to simplify the expression inside the integral.

Let \(u = 7x + 5\). Then, compute the differential \(du = 7 \, dx\), which implies \(dx = \frac{du}{7}\).

Express \(x\) in terms of \(u\) by rearranging the substitution: \(x = \frac{u - 5}{7}\).

Rewrite the integral entirely in terms of \(u\): replace \(x\) with \(\frac{u - 5}{7}\) and \(dx\) with \(\frac{du}{7}\), so the integral becomes \(\int \left( \frac{u - 5}{7} \right) u^{3/2} \cdot \frac{du}{7}\).

Simplify the constants and the integrand, then use the table of integrals to find the integral of the resulting expression in terms of \(u\). After integrating, substitute back \(u = 7x + 5\) to express the answer in terms of \(x\).

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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 (7x + 5)^(3/2).

Using a Table of Integrals

A table of integrals provides formulas for common integral forms, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form and matching it to an entry in the table can save time and reduce errors. It is important to adjust the integral to fit the table's format, often using substitution.

Power Functions and Exponents in Integration

Integrating functions with fractional exponents, like (7x + 5)^(3/2), requires understanding how to handle power functions. The power rule for integration extends to fractional powers, where the integral of x^n is (x^(n+1))/(n+1) plus a constant, provided n ≠ -1. Recognizing this helps in applying the correct formula or substitution.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ x (7x + 5)^(3/2) dx