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Chapter 4: Problem 41

Determine these indefinite integrals. $$\int(3 x+2)^{2} d x\( \)(Hint: Expand first.)$

Short Answer

Expert verified
The integral of \( (3x + 2)^2 dx = 3x^3 + 6x^2 + 4x + C \.

Step by step solution

01

- Expand the binomial

Expand the expression \(3x + 2\)^2 using the binomial theorem: \[ (3x + 2)^2 = (3x)^2 + 2 \cdot 3x \cdot 2 + 2^2 = 9x^2 + 12x + 4. \] This gives us the expanded form before integrating.
02

- Write the integral with the expanded form

Substitute the expanded form back into the integral: \[ \int(3x + 2)^2 dx = \int(9x^2 + 12x + 4) dx \]
03

- Integrate term by term

Now integrate each term separately: \[ \int 9x^2 dx + \int 12x dx + \int 4 dx. \] The integrations are as follows: \[ \int 9x^2 dx = 9 \int x^2 dx = 9 \frac{x^3}{3} = 3x^3, \int 12x dx = 12 \int x dx = 12 \frac{x^2}{2} = 6x^2, \int 4 dx = 4x. \]
04

- Combine the integrated terms and include the constant of integration

Add all the integrated terms together and include the constant of integration \(C\): \[ 3x^3 + 6x^2 + 4x + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant of Integration
Finally, it鈥檚 crucial to include the constant of integration, \C\.
This constant appears because an indefinite integral represents a family of functions differing only by a constant.
Thus, after integrating term-by-term, we combine all the terms:
\(3x^3 + 6x^2 + 4x + C\).
Neglecting \C\ would suggest a single specific solution, but there are infinitely many solutions differing by a constant.

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