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Chapter 7: Problem 46

Evaluate the following integrals or state that they diverge. $$\int_{1}^{11} \frac{d x}{(x-3)^{2 / 3}}$$

Short Answer

Expert verified
Question: Evaluate the integral of the function, or determine if it diverges: $$\int_{1}^{11} \frac{d x}{(x-3)^{2 / 3}} $$ Answer: The evaluated integral is \(6 - 3(-2)^{1/3}\).

Step by step solution

01

Setting up the indefinite integral

First, we need to find the indefinite integral of the given function: $$\int \frac{d x}{(x-3)^{2 / 3}}$$
02

Use substitution

We can apply the substitution method. Let \(u = x - 3\). Therefore, \(d x = d u\). Now the integral becomes: $$\int \frac{d u}{u^{2 / 3}}$$
03

Rewrite the integral as a power function

Rewrite the integral using the power rule formula (i.e., \(\int x^n dx\)). To do this, we can write \(\frac{1}{u^{2/3}}\) as \(u^{-2/3}\). $$\int u^{- 2 / 3} d u$$
04

Apply the Power Rule

The power rule states that \(\int x^n dx = \frac{1}{n+1} x^{n+1} + C\). Apply the power rule to our integral: $$\frac{1}{(-2/3)+1} u^{(-2/3)+1} + C$$
05

Simplify the result

Simplify the expression: $$\frac{3}{1} u^{1/3} + C = 3u^{1/3} + C$$
06

Substitute x back in

Replace \(u\) with \(x - 3\) to obtain the result of the indefinite integral: $$3(x - 3)^{1/3} + C$$
07

Evaluate the definite integral

To evaluate the definite integral, simply substitute the limits of integration (1 and 11) into the indefinite integral and subtract the values: $$\int_{1}^{11} \frac{d x}{(x-3)^{2 / 3}} = \left[3(x - 3)^{1/3}\right]_{1}^{11} = 3(11 - 3)^{1/3} - 3 (1 - 3)^{1/3}$$
08

Simplify the result

Simplify the expression: $$3(8)^{1/3} - 3(-2)^{1/3} = 3 \cdot 2 - 3(-2)^{1/3} = 6 - 3(-2)^{1/3}$$ The evaluated integral is \(6 - 3(-2)^{1/3}\).

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Most popular questions from this chapter

Evaluate the following integrals. Consider completing the square. $$\int_{2+\sqrt{2}}^{4} \frac{d x}{\sqrt{(x-1)(x-3)}}$$

a. Graph the functions \(f_{1}(x)=\sin ^{2} x\) and \(f_{2}(x)=\sin ^{2} 2 x\) on the interval \([0, \pi] .\) Find the area under these curves on \([0, \pi]\) b. Graph a few more of the functions \(f_{n}(x)=\sin ^{2} n x\) on the interval \([0, \pi],\) where \(n\) is a positive integer. Find the area under these curves on \([0, \pi] .\) Comment on your observations. c. Prove that \(\int_{0}^{\pi} \sin ^{2}(n x) d x\) has the same value for all positive integers \(n\) d. Does the conclusion of part (c) hold if sine is replaced by cosine? e. Repeat parts (a), (b), and (c) with \(\sin ^{2} x\) replaced by \(\sin ^{4} x\) Comment on your observations. f. Challenge problem: Show that, for \(m=1,2,3, \ldots\) $$\int_{0}^{\pi} \sin ^{2 m} x d x=\int_{0}^{\pi} \cos ^{2 m} x d x=\pi \cdot \frac{1 \cdot 3 \cdot 5 \cdots(2 m-1)}{2 \cdot 4 \cdot 6 \cdots 2 m}$$

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{x^{3}+1}{x\left(x^{2}+x+1\right)^{2}} d x$$

a. Verify the identity \(\sec x=\frac{\cos x}{1-\sin ^{2} x}\) b. Use the identity in part (a) to verify that \(\int \sec x d x=\frac{1}{2} \ln \left|\frac{1+\sin x}{1-\sin x}\right|+C\) (Source: The College Mathematics Joumal \(32,\) No. 5 (November 2001))

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{\left(e^{x}+e^{-x}\right)^{2}}$$
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