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Chapter 4: Problem 7

Find the indefinite integral and check the result by differentiation. $$ \int(1+2 x)^{4}(2) d x $$

Short Answer

Expert verified
The indefinte integral of \((1+2x)^4(2)\) is \(\frac{1}{5} (1+2x)^5 + C\), where \(C\) is the constant of integration.

Step by step solution

01

Apply Substitution Rule for Integration

Choose \(u = 1 + 2x\), hence \(du = 2dx\). Thus, the integral transforms into \(\int u^4 du\).
02

Calculate Integrals Using Power Rule

The power rule for integration states, when integrating a function \(u^n\), where \(n \neq -1\), the result is \(\frac{1}{n+1}u^{n+1} + C\), where \(C\) is the constant of integration. In this case, the integral of \(u^4\) w.r.t. \(u\) becomes: \(\frac{1}{5} u^5 + C\).
03

Substitute Back Original Variable

Now, we replace \(u\) with our original expression: \(\frac{1}{5} (1+2x)^5 + C\)
04

Verify Result through Differentiation

To verify if this result is correct, derive it w.r.t. \(x\). Make use of the chain rule \((f(g(x)))' = f'(g(x)) . g'(x)\). This gives us back the original integrand: \((1+2x)^4(2)\)

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Most popular questions from this chapter

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t $$

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