- Solve the substitution \(u = x - 4\) for \(x\) to get \(x = u + 4\) and note that \(dx = du\).
- Rewrite the integral \(\int \frac{x}{\sqrt{x-4}} \, dx\) as \(\int \frac{u+4}{\sqrt{u}} \, du\).
- Simplify the integrand to \(u^{1/2} + 4u^{-1/2}\) and integrate using the power rule to obtain \(\frac{2}{3}u^{3/2} + 8u^{1/2} + C\).
- Substitute \(u = x - 4\) back into the result to find the final integral: \(\boxed{\frac{2}{3}(x-4)^{3/2} + 8\sqrt{x-4} + C}\).
Explanation
- Problem Analysis and Substitution Setup
The goal is to evaluate the indefinite integral \(\int \frac{x}{\sqrt{x-4}} \, dx\) using the method of u-substitution. We are given the substitution \(u = x - 4\). We must first express all variables in the integral in terms of \(u\), integrate, and then convert back to \(x\).
- Substitution and Rewriting the Integral
First, we solve the given substitution \(u = x - 4\) for \(x\), which gives:
\[ x = u + 4 \]
Next, we find the differential \(du\). Taking the derivative of \(u = x - 4\) with respect to \(x\) gives \(\frac{du}{dx} = 1\), which implies:
\[ du = dx \]
Now we substitute \(x = u + 4\) and \(dx = du\) into the original integral:
\[ \int \frac{x}{\sqrt{x-4}} \, dx = \int \frac{u+4}{\sqrt{u}} \, du \]
- Evaluating the Integral
Now we simplify the integrand \(\frac{u+4}{\sqrt{u}}\) by splitting the fraction:
\[ \int \left( \frac{u}{\sqrt{u}} + \frac{4}{\sqrt{u}} \right) \, du = \int (u^{1 - 1/2} + 4u^{-1/2}) \, du = \int (u^{1/2} + 4u^{-1/2}) \, du \]
Applying the power rule for integration \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\) (where \(n
eq -1\)):
\[ \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3}u^{3/2} \]
\[ \int 4u^{-1/2} \, du = 4 \left( \frac{u^{1/2}}{1/2} \right) = 8u^{1/2} \]
Combining these, we get:
\[ \frac{2}{3}u^{3/2} + 8u^{1/2} + C \]
- Final Answer and Back-Substitution
Finally, substitute \(u = x - 4\) back into the expression:
\[ \frac{2}{3}(x-4)^{3/2} + 8(x-4)^{1/2} + C \]
This can be factored further or left in this form:
\[ \frac{2}{3}(x-4)^{1/2} (x-4 + 12) + C = \frac{2}{3}(x-4)^{1/2} (x+8) + C \]
The integral evaluates to \(\boxed{\frac{2}{3}(x-4)^{3/2} + 8\sqrt{x-4} + C}\).
Examples
In fields like fluid dynamics or engineering, you often need to calculate work done by pressure or force over a distance. If the force applied varies with the displacement in a non-linear way, such as \(F(x) = \frac{x}{\sqrt{x-4}}\), integration is essential to find the total energy or work. Using techniques like substitution helps simplify complex expressions into solvable forms, allowing for precise design and performance analysis.