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Chapter 7: Problem 38

Evaluate the following integrals or state that they diverge. $$\int_{-3}^{1} \frac{d x}{(2 x+6)^{2 / 3}}$$

Short Answer

Expert verified
Question: Evaluate the definite integral: $$\int_{-3}^{1} \frac{d x}{(2 x+6)^{2 / 3}}$$ Answer: The definite integral evaluates to 3.

Step by step solution

01

Simplify the given function

First, we can simplify the given function by factoring out the constant from the denominator: $$\int_{-3}^{1} \frac{d x}{(2 x+6)^{2 / 3}} = \int_{-3}^{1} \frac{d x}{(2(x+3))^{2 / 3}}$$
02

Rewrite the function

Next, we can rewrite the function as follows: $$\int_{-3}^{1} \frac{d x}{(2(x+3))^{2 / 3}} = \frac{1}{2^{2/3}} \int_{-3}^{1} (x+3)^{-2 / 3} dx$$
03

Find the antiderivative of the function

Now we need to find the antiderivative of the function \((x+3)^{-2/3}\). Applying the power rule for integration: $$\int (x+3)^{-2/3} dx = \frac{(x+3)^{1/3}}{1/3} + C = 3(x+3)^{1/3} + C$$
04

Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative, we can apply the Fundamental Theorem of Calculus to find the value of the definite integral: $$\frac{1}{2^{2/3}} \int_{-3}^{1} (x+3)^{-2 / 3} dx = \frac{1}{2^{2/3}} \left[3(x+3)^{1/3}\right]_{-3}^{1}$$
05

Evaluate the definite integral

Finally, we evaluate the definite integral by plugging the limits of integration into the antiderivative: $$\frac{1}{2^{2/3}} \left[3(x+3)^{1/3}\right]_{-3}^{1} = \frac{1}{2^{2/3}} \left[3(4)^{1/3} - 3(0)^{1/3}\right] = \frac{1}{2^{2/3}}[3(2^{\frac{2}{3}})] = 3$$ Thus, the value of the given integral is: $$\int_{-3}^{1} \frac{d x}{(2 x+6)^{2 / 3}} = 3$$

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Most popular questions from this chapter

Determine whether the following statements are true and give an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of 16.

Use integration by parts to evaluate the following integrals. $$\int_{0}^{1} x \ln x d x$$

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Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{2 \pi} \frac{d x}{(5+3 \sin x)^{2}}=\frac{5 \pi}{32}\)

A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=\sin y$$
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