Problem 7
Question
Find the indefinite integral and check the result by differentiation. $$ \int(1+2 x)^{4}(2) d x $$
Step-by-Step Solution
Verified Answer
The indefinte integral of \((1+2x)^4(2)\) is \(\frac{1}{5} (1+2x)^5 + C\), where \(C\) is the constant of integration.
1Step 1: Apply Substitution Rule for Integration
Choose \(u = 1 + 2x\), hence \(du = 2dx\). Thus, the integral transforms into \(\int u^4 du\).
2Step 2: 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: \(\frac{1}{5} u^5 + C\).
3Step 3: Substitute Back Original Variable
Now, we replace \(u\) with our original expression: \(\frac{1}{5} (1+2x)^5 + C\)
4Step 4: 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: \((1+2x)^4(2)\)
Other exercises in this chapter
Problem 7
Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{-1}^{0}(x-2) d x $$
View solution Problem 7
In Exercises 3-8, evaluate the definite integral by the limit definition. $$ \int_{1}^{2}\left(x^{2}+1\right) d x $$
View solution Problem 7
Find the indefinite integral. $$ \int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x} d x $$
View solution Problem 7
In Exercises 7 -12, use sigma notation to write the sum. $$ \frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)} $$
View solution