Question Statement

Evaluate the integral:

$\int \frac{\ln x}{\sqrt{x}}dx$

---

Background and Explanation

This integral involves the product of a logarithmic function ($\ln x$) and a power function ($x^{-1/2}$). Since direct integration is not possible here, we use integration by parts, which is the product rule for integration. The LIATE rule suggests choosing the logarithmic function as $u$ since it simplifies when differentiated.

---

Solution

First, rewrite the integrand to make the power explicit:

$I=\int \frac{\ln x}{\sqrt{x}}dx=\int \ln x\cdot x^{-1/2}dx$

Step 1: Set up Integration by Parts

Using the formula $\int udv=uv-\int vdu$, we choose:

• $u=\ln x$ (so that $du=\frac{1}{x}dx$)
• $dv=x^{-1/2}dx$ (so we need to find $v$)

Step 2: Find $v$ by integrating $dv$

$v=\int x^{-1/2}dx=\frac{x^{-1/2+1}}{-1/2+1}=\frac{x^{1/2}}{1/2}=2\sqrt{x}$

Step 3: Apply the Integration by Parts Formula

Substituting into the formula:

$\begin{array}{rl}I& =\ln x\cdot (2\sqrt{x})-\int (2\sqrt{x})\cdot \frac{1}{x}dx\\ & =2\sqrt{x}\ln x-2\int \frac{x^{1/2}}{x}dx\\ & =2\sqrt{x}\ln x-2\int x^{1/2-1}dx\\ & =2\sqrt{x}\ln x-2\int x^{-1/2}dx\end{array}$

Step 4: Evaluate the Remaining Integral

The remaining integral is the same form we started with for $v$:

$\int x^{-1/2}dx=\frac{x^{1/2}}{1/2}=2\sqrt{x}$

Substituting back:

$\begin{array}{rl}I& =2\sqrt{x}\ln x-2(2\sqrt{x})+c\\ & =2\sqrt{x}\ln x-4\sqrt{x}+c\end{array}$

Step 5: Factor the Result

Factor out the common term $2\sqrt{x}$:

$I=2\sqrt{x}(\ln x-2)+c$

---

Key Formulas or Methods Used

• Integration by Parts: $\int udv=uv-\int vdu$
• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$ (for $n\ne -1$)
• Derivative of Natural Logarithm: $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• Exponent Rules: $\frac{1}{\sqrt{x}}=x^{-1/2}$ and $\frac{x^a}{x^b}=x^{a-b}$

---

Summary of Steps

1. Rewrite $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$ to identify the power function
2. Choose $u=\ln x$ and $dv=x^{-1/2}dx$ (following LIATE priority)
3. Compute $du=\frac{1}{x}dx$ and $v=2\sqrt{x}$
4. Apply integration by parts: $uv-\int vdu$
5. Simplify the new integral to $2\int x^{-1/2}dx$
6. Evaluate to get $4\sqrt{x}$
7. Combine terms and factor out $2\sqrt{x}$ for the final answer: $2\sqrt{x}(\ln x-2)+c$