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Chapter 17: Problem 5

Find \(\int(x+3)^{2} \mathrm{~d} x\) (be careful!).

Short Answer

Expert verified
Question: Find the integral of (x+3)^2 with respect to x. Answer: The integral of (x+3)^2 with respect to x is (∫(x+3)^2 dx = (1/3)x^3 + 3x^2 + 9x + C).

Step by step solution

01

Identify the function to be integrated

We are given the function \((x+3)^2\). Our goal is to find the integral, i.e., the antiderivative of this function with respect to x: \(\int(x+3)^2 \mathrm{~d} x\).
02

Apply the power rule for integration

We will apply the power rule for integration, which states that if we want to find the integral of \(x^n\), it would be \(\frac{1}{n+1}x^{n+1} + C\). However, since we have more complicated function \((x+3)^2\), we will first simplify it before applying the power rule. Simplifying \((x+3)^2\), we get: \((x+3)^2 = x^2 + 6x + 9\)
03

Integrate term-by-term

Now that we have the expanded form of \((x+3)^2\), we can integrate it term-by-term with respect to x: \(\int(x^2 + 6x + 9) \mathrm{~d} x = \int x^2 \mathrm{~d} x + \int 6x \mathrm{~d} x + \int 9 \mathrm{~d} x\)
04

Apply the power rule to each term

Next, we will apply the power rule to each term in the integral: 1. For \(\int x^2 \mathrm{~d} x\), we apply the power rule with \(n=2\) and we get \(\frac{1}{3}x^3\). 2. For \(\int 6x \mathrm{~d} x\), we apply the power rule with \(n=1\) and we get \(3x^2\). 3. For \(\int 9 \mathrm{~d} x\), since there's no x term, we can just multiply the constant (9) with x to get \(9x\).
05

Combine the results and add the constant of integration

Now, we combine the results from each term and add the constant of integration (C): \(\int(x+3)^2 \mathrm{~d} x = \frac{1}{3}x^3 + 3x^2 + 9x + C\) So, the integral of \((x+3)^2\) with respect to x is: \(\int(x+3)^2 \mathrm{~d} x = \frac{1}{3}x^3 + 3x^2 + 9x + C\).

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