Question

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

Short answer

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

Step-by-step solution

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.

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.

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, \( 2 (1+2x)^4 \).