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Chapter 7: Problem 1

What change of variables would you use for the integral \(\int(4-7 x)^{-6} d x ?\)

Short Answer

Expert verified
Answer: -\(\frac{1}{7} \int u^{-6} du\)

Step by step solution

01

Choose a change of variables

Let's use a change of variables \(u = 4 - 7x\). This linear function appears in the integrand, making a substitution with it likely to simplify the expression.
02

Substitute and Find du

Now that we have chosen a change of variables \(u = 4 - 7x\), we need to find \(\frac{du}{dx}\) and substitute it back into the integral. Solving for \(\frac{du}{dx}\), we get: \[\frac{du}{dx} = -7\] Now multiply both sides by \(dx\) and rearrange: \[du = -7 dx\] \[dx = -\frac{1}{7} du\]
03

Modify the integral using the change of variables

Using the change of variables \(u = 4 - 7x\) and \(dx = -\frac{1}{7} du\), we can rewrite the original integral as follows: \[\int (4 - 7x)^{-6} dx = \int u^{-6} \cdot -\frac{1}{7} du\]
04

Simplify the new integral

Now, let's simplify the result from Step 3 by taking out the constant factor: \[\int u^{-6} \cdot -\frac{1}{7} du = -\frac{1}{7} \int u^{-6} du\] After the change of variables and necessary simplification, we are left with a much simpler integral to evaluate: \[-\frac{1}{7} \int u^{-6} du\]

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