Integrate (2x+1)$^3$

To solve the integral of $(2x + 1)^3$ with respect to $x$, we use the method of Integration by Substitution (or the reverse chain rule).

1. Analyze the question

We are asked to find the indefinite integral:

2. Explain the concept

When integrating a function of the form $(ax + b)^n$, where $a$ and $b$ are constants, we use the rule:

This rule works because we increase the power by 1 and then divide by the new power, while also dividing by the derivative of the inner function $(2x + 1)$, which is $2$.

3. Show the solution

• Step 1: Identify the components. Here, $n = 3$ and the derivative of the inner function $(2x + 1)$ is $2$.
• Step 2: Apply the power rule for integration. Increase the exponent by 1: $$(2x + 1)^{3+1} = (2x + 1)^4$$
• Step 3: Divide by the new exponent ($4$) and multiply by the derivative of the inner function ($2$): $$\text{Result} = \frac{(2x + 1)^4}{4 \times 2} + C$$
• Step 4: Simplify the denominator: $$\text{Result} = \frac{(2x + 1)^4}{8} + C$$

4. Identify the answer

Comparing our result to the given options, we find that it matches option B.

5. Exam Tip

Whenever you see a linear expression inside a power (like $ax+b$), don't forget to divide by the coefficient of $x$ (the $a$ value). Many students forget this step and choose the wrong answer. You can always check your answer by differentiating it—if you differentiate $\frac{(2x+1)^4}{8} + C$, you will get back to $(2x+1)^3$ using the chain rule!