To solve the given integral:

$\int (1 + 2x)^{22} \, dx$

We can use substitution to simplify the expression. Here's the step-by-step solution:

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Step 1: Substitution

Let: $u = 1 + 2x \quad \text{so that} \quad \frac{du}{dx} = 2 \quad \text{or} \quad dx = \frac{du}{2}.$

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Step 2: Rewrite the integral

Substitute $u = 1 + 2x$ and $dx = \frac{du}{2}$:

$\int (1 + 2x)^{22} \, dx = \int u^{22} \cdot \frac{du}{2}.$

Factor out the constant $\frac{1}{2}$:

$\frac{1}{2} \int u^{22} \, du.$

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Step 3: Apply the power rule for integration

The power rule states: $\int u^n \, du = \frac{u^{n+1}}{n+1} + C.$

Here, $n = 22$. Apply the rule:

$\frac{1}{2} \cdot \frac{u^{22+1}}{22+1} + C = \frac{1}{2} \cdot \frac{u^{23}}{23} + C = \frac{u^{23}}{46} + C.$

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Step 4: Back-substitute $u = 1 + 2x$

Replace $u$ with $1 + 2x$:

$\int (1 + 2x)^{22} \, dx = \frac{(1 + 2x)^{23}}{46} + C.$

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Final Answer:

$\boxed{\int (1 + 2x)^{22} \, dx = \frac{(1 + 2x)^{23}}{46} + C.}$

Would you like further details or a verification? Let me know!

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Related Questions:

1. How do you apply substitution for integrals involving powers of binomials?
2. Can this integral be expanded using the binomial theorem instead of substitution?
3. How does substitution help simplify complex integrals?
4. How can we compute definite integrals for expressions like this one?
5. What are other methods to solve integrals involving exponential terms?

Tip:

Always check if substitution simplifies your integral before attempting more advanced techniques.