Problem 9 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The indefinite integral of the function is \( \frac{1}{5}(1+2x)^5 + C \).

Step 1: Perform Integration

The given integral can be written in a more digestible format as: \( \int 2(1+2x)^4 dx \). In order to perform this integration, a substitution method can be used. Let the substitution \( u = 1 + 2x \). This gives \( du = 2dx \). Therefore, the integral becomes: \( \int u^4 du \). Solving this gives the integrated function as \( \frac{1}{5}u^5 + C \), where C is the constant of integration.

Step 2: Substitute Back

Now, substituting u back in terms of x, the solution becomes \( \frac{1}{5}(1+2x)^5 + C \). This is the indefinite integral of the given function.

Step 3: Check the Result by Differentiation

Now, let's ensure correctness by differentiating the obtained function. The rule of differentiation states that the derivative of an indefinite integral of a function is the original function. Hence, differentiate \( \frac{1}{5}(1+2x)^5 \) using the power and chain rule, and this should give the original function, \( 2 (1+2x)^4 \).