To solve the integral of \((t - 2)\sqrt{1 + t} \, dt\), we can approach it step-by-step.

\[
I = \int (t - 2)\sqrt{1 + t} \, dt
\]

### Step 1: Break the integral into two parts:
We distribute the term \(\sqrt{1 + t}\):

\[
I = \int t \sqrt{1 + t} \, dt - 2 \int \sqrt{1 + t} \, dt
\]

### Step 2: Solve each integral separately.

#### Part 1: \(\int t \sqrt{1 + t} \, dt\)

Let’s use substitution. Set \(u = 1 + t\), which gives \(du = dt\) and \(t = u - 1\).

Now the integral becomes:

\[
\int t \sqrt{1 + t} \, dt = \int (u - 1) \sqrt{u} \, du = \int (u^{3/2} - u^{1/2}) \, du
\]

Now integrate term by term:

\[
\int u^{3/2} \, du = \frac{2}{5} u^{5/2}, \quad \int u^{1/2} \, du = \frac{2}{3} u^{3/2}
\]

So,

\[
\int (u^{3/2} - u^{1/2}) \, du = \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2}
\]

Substitute \(u = 1 + t\) back:

\[
\int t \sqrt{1 + t} \, dt = \frac{2}{5} (1 + t)^{5/2} - \frac{2}{3} (1 + t)^{3/2}
\]

#### Part 2: \(-2 \int \sqrt{1 + t} \, dt\)

Again, using substitution \(u = 1 + t\), \(du = dt\), we get:

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

Substitute \(u = 1 + t\) back:

\[
-2 \int \sqrt{1 + t} \, dt = -\frac{4}{3} (1 + t)^{3/2}
\]

### Step 3: Combine the results

Now, we combine the two parts:

\[
I = \left( \frac{2}{5} (1 + t)^{5/2} - \frac{2}{3} (1 + t)^{3/2} \right) - \frac{4}{3} (1 + t)^{3/2}
\]

Simplify the expression:

\[
I = \frac{2}{5} (1 + t)^{5/2} - \frac{2}{3} (1 + t)^{3/2} - \frac{4}{3} (1 + t)^{3/2}
\]
\[
I = \frac{2}{5} (1 + t)^{5/2} - 2 (1 + t)^{3/2}
\]

Thus, the integral of \((t - 2)\sqrt{1 + t} \, dt\) is:

\[
I = \frac{2}{5} (1 + t)^{5/2} - 2 (1 + t)^{3/2} + C
\]

where \(C\) is the constant of integration.

---

Would you like any further details or clarification on any step?

Here are some related questions you might explore:
1. How do substitutions simplify integrals with radicals?
2. What other methods can be used to integrate functions like this?
3. How can we visualize the integral geometrically?
4. Can this result be applied to solve any differential equations?
5. How do you confirm the result of an integral through differentiation?

**Tip:** Always check your integration by differentiating the result to see if you get back the original function.

Learn how to solve the integral of (t - 2)√(1 + t) step-by-step using substitution. This solution involves distributing radicals and integrating using the substitution method, perfect for undergraduate calculus students.

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