Problem 1 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Find the solution for the integral ∫(4-7x)^-6 dx using substitution. Answer: The solution for the integral ∫(4-7x)^-6 dx is -1/35(4-7x)^{-5} + C.

Step 1: Identify the Substitution

We will replace the expression inside the parentheses with a new variable such that the integral becomes easier to calculate. In this case, a suitable substitution would be: $$u = 4 - 7x$$

Step 2: Find the Differential of the New Variable

Now that we have defined the substitution, let's find the differential \(du\): $$du = -7 dx$$ We also need to find the corresponding value of \(dx\): $$dx = \frac{-1}{7} du$$

Step 3: Substitute the Variable and Differential

Now we will replace all the \(x\) terms and the \(dx\) term with the new variable \(u\) and its corresponding differential: $$\int(4-7x)^{-6} dx = \int u^{-6} (\frac{-1}{7} du)$$

Step 4: Integrate with the New Variable

Now we can integrate the function with respect to the new variable \(u\): $$\int u^{-6} (\frac{-1}{7} du) = \frac{-1}{7}\int u^{-6} du $$ Now, we can integrate the function using the power rule: $$\frac{-1}{7}\int u^{-6} du = \frac{-1}{7} \cdot \frac{u^{-5}}{-5} + C$$ _where C is the constant of integration._

Step 5: Replace the New Variable with the Original Variable

Finally, we replace the new variable \(u\) with the original variable \(x\) by using the substitution we defined earlier: $$\frac{-1}{7} \cdot \frac{u^{-5}}{-5} + C = \frac{-1}{7} \cdot \frac{(4-7x)^{-5}}{-5} + C$$ And we simplify the expression: $$-\frac{1}{35} (4-7x)^{-5} + C$$ Therefore, the final solution for the integral is: $$\int(4-7x)^{-6} dx = -\frac{1}{35} (4-7x)^{-5} + C$$