Chapter 5: Q. 29 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x$

Short Answer

Expert verified

The value of the integral is $\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x = 1 + \sqrt[3]{2}$

Step by step solution

Step 1. Given information.

The given integral is the following.

$\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x$

Step 2. Value of integral.

The function $f (x) = {(3 x - 1)}^{\frac{- 2}{3}}$ has a vertical asymptote at $x = \frac{1}{3} .$

solve this improper integral by splitting at localid="1648875492622" $x = \frac{1}{3} .$

localid="1648876694124" $\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x = \int_{0}^{\frac{1}{3}} {(3 x - 1)}^{\frac{- 2}{3}} d x + \int_{\frac{1}{3}}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x = \lim_{b \to {\frac{1}{3}}^{-}} \int_{0}^{b} {(3 x - 1)}^{\frac{- 2}{3}} d x + \lim_{a \to {\frac{1}{3}}^{+}} \int_{a}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x = \lim_{b \to {\frac{1}{3}}^{-}} {[{(3 x - 1)}^{\frac{1}{3}}]}_{0}^{b} + \lim_{a \to {\frac{1}{3}}^{+}} {[{(3 x - 1)}^{\frac{1}{3}}]}_{a}^{1} = \lim_{b \to {\frac{1}{3}}^{-}} ({(3 b - 1)}^{\frac{1}{3}} - \sqrt[3]{- 1}) + \lim_{a \to {\frac{1}{3}}^{+}} (\sqrt[3]{2} - {(3 a - 1)}^{\frac{1}{3}}) = 0 + 1 + \sqrt[3]{2} - 0 = 1 + \sqrt[3]{2}$

So the value of the integral is localid="1648876733115" $\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x = 1 + \sqrt[3]{2} .$

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Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

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Use limits of definite integrals to calculate each of the improper integrals in Exercises.∫013x-1-23dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

One App. One Place for Learning.

Most popular questions from this chapter

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Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{0}^{1} {(3 x - 1)}^{\frac{- 2}{3}} d x$