https://r2.erweima.ai/mathbot/image/1725838862191hjm4qtkg.jpg

The integral shown in the image is:

\[
\int \frac{-3x^2}{\sqrt{x^3 - 4}} \, dx
\]

To solve this integral, we can use a substitution method. Let's proceed step by step:

### Step 1: Substitution
Let:
\[
u = x^3 - 4
\]

Then, the derivative of \( u \) with respect to \( x \) is:

\[
\frac{du}{dx} = 3x^2 \implies du = 3x^2 \, dx
\]

Now, rewrite the integral in terms of \( u \):

\[
dx = \frac{du}{3x^2}
\]

Substituting into the integral:

\[
\int \frac{-3x^2}{\sqrt{x^3 - 4}} \, dx = \int \frac{-3x^2}{\sqrt{u}} \cdot \frac{du}{3x^2} = \int \frac{-1}{\sqrt{u}} \, du
\]

### Step 2: Simplifying the Integral
The integral simplifies to:

\[
\int -u^{-\frac{1}{2}} \, du = -2u^{\frac{1}{2}} + C = -2\sqrt{u} + C
\]

### Step 3: Substitute \( u \) Back in Terms of \( x \)
Replace \( u \) with \( x^3 - 4 \):

\[
-2\sqrt{x^3 - 4} + C
\]

So, the solution to the integral is:

\[
\int \frac{-3x^2}{\sqrt{x^3 - 4}} \, dx = -2\sqrt{x^3 - 4} + C
\]

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How would the integration change if the numerator were a different function of \(x\)?
2. What if the denominator had a cubic root instead of a square root?
3. How does substitution work when integrating trigonometric functions?
4. Can this substitution method be used for definite integrals?
5. How does this solution relate to solving differential equations?

**Tip:** Always check the substitution and back-substitution steps carefully to ensure they simplify correctly.

Evaluate the integral of (-3x^2)/(sqrt(x^3 - 4)) dx

Step-by-step solution to the integral (-3x²)/(√(x³ - 4)) using substitution. Learn how to apply the substitution method, simplify the integral, and back-substitute to get the final answer. Suitable for college-level calculus students.

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