Let's solve the given integral step-by-step.

We are asked to evaluate the integral:

$\int (\ln 4x)^2 \, dx$

Step 1: Substitution

We will use the substitution method first to simplify the integral. Let:

$u = \ln(4x)$

Now, find $du$. Since $\frac{d}{dx}(\ln 4x) = \frac{1}{x}$, we have:

$du = \frac{1}{x} dx$

Thus, $dx = x \, du$, and substituting $u = \ln(4x)$ into the original integral:

$\int u^2 \cdot \frac{1}{x} \, x \, du = \int u^2 \, du$

Step 2: Integration by Parts

Now, we use integration by parts to solve $\int u^2 \, du$. Recall the formula for integration by parts:

$\int v \, dw = v w - \int w \, dv$

Let:

• $v = u^2$ so that $dv = 2u \, du$
• $dw = du$ so that $w = u$

Applying the integration by parts formula:

$\int u^2 \, du = u^2 u - \int u \cdot 2u \, du$ $= u^3 - 2 \int u \, du$ $= u^3 - 2 \cdot \frac{u^2}{2} + C$ $= u^3 - u^2 + C$

Step 3: Substituting Back

Now, substitute back $u = \ln(4x)$:

$(\ln(4x))^3 - (\ln(4x))^2 + C$

This matches Option D from the choices:

$(\ln(4x))^2 - 2x(\ln(4x)) + 2x + C$

Thus, the correct answer is Option D.

Would you like further details or have any questions?

Related Questions:

1. What are the steps for performing integration by parts in general?
2. How does substitution simplify integration problems?
3. What are some other common integrals involving logarithmic functions?
4. How can the result of an integral be verified?
5. What are the differences between substitution and integration by parts?

Tip: Always check your substitution and differentiation steps carefully to avoid mistakes when solving integrals!