To find the antiderivative (or primitive) of the function \( f(x) = (x+2)\sqrt{3x+6} \), we can proceed using the method of substitution. Here's the step-by-step solution:

### 1. Substitution

Let \( u = 3x + 6 \). Then, the derivative of \( u \) with respect to \( x \) is:

\[
\frac{du}{dx} = 3 \implies dx = \frac{du}{3}
\]

We can express \( x \) in terms of \( u \):

\[
u = 3x + 6 \implies x = \frac{u - 6}{3}
\]

### 2. Express the Function in Terms of \( u \)

The original function \( (x+2)\sqrt{3x+6} \) becomes:

\[
(x+2)\sqrt{3x+6} = \left(\frac{u - 6}{3} + 2\right)\sqrt{u}
\]

Simplify the expression inside the parentheses:

\[
\frac{u - 6}{3} + 2 = \frac{u - 6 + 6}{3} = \frac{u}{3}
\]

So, the integrand becomes:

\[
\frac{u}{3} \sqrt{u} = \frac{u^{3/2}}{3}
\]

### 3. Set Up the Integral

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

\[
\int (x+2)\sqrt{3x+6} \, dx = \int \frac{u^{3/2}}{3} \cdot \frac{du}{3} = \frac{1}{9} \int u^{3/2} \, du
\]

### 4. Integrate Using the Power Rule

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

Therefore, the integral is:

\[
\frac{1}{9} \cdot \frac{2}{5} u^{5/2} = \frac{2}{45} u^{5/2}
\]

### 5. Back-Substitute \( u \)

Recall that \( u = 3x + 6 \):

\[
\frac{2}{45} (3x+6)^{5/2}
\]

### 6. Add the Constant of Integration

\[
\int (x+2)\sqrt{3x+6} \, dx = \frac{2}{45} (3x+6)^{5/2} + C
\]

So, the antiderivative of \( (x+2)\sqrt{3x+6} \) is \( \frac{2}{45} (3x+6)^{5/2} + C \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you choose the right substitution when integrating?
2. Can integration by parts be used for this integral? If so, how?
3. What is the relationship between differentiation and integration?
4. How would you compute the definite integral of this function over a specific interval?
5. How does the method of substitution work for trigonometric functions?

**Tip**: Always remember to add the constant of integration \( C \) when computing indefinite integrals.

Learn how to find the antiderivative (primitive) of the function (x+2)*sqrt(3x+6) using the substitution method. This detailed solution covers the steps involved in integrating the function, including substitution and integration by parts considerations. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation