91Ó°ÊÓ

Integration techniques are crucial for finding the antiderivative of a given function. The method generally involves reversing the process of differentiation. In our exercise, the function given is a sum: \( f(u) = \frac{2}{u} + u \). To tackle this, we take the antiderivative of each part separately.

• The term \( \frac{2}{u} \) can be integrated as \( 2 \ln |u| \). This is because the derivative of \( \ln |u| \) is \( \frac{1}{u} \), so scaling by 2 gives us \( \frac{2}{u} \).
• For the linear term \( u \), we apply the power rule for integration. Increasing the exponent by one and dividing by the new exponent, we find that the integral is \( \frac{u^2}{2} \).

By following the steps for these two components carefully, we combine them to form the general antiderivative: \[ F(u) = 2 \ln |u| + \frac{u^2}{2} + C \]where \( C \) is an arbitrary constant that we'll determine using additional information.

Initial conditions are vital for finding a specific antiderivative from a general solution. They provide us with the information needed to pinpoint the exact function among infinitely many solutions. In our example, we're given the condition \( F(1) = 5 \).This condition tells us that when \( u = 1 \), the function \( F \) results in 5. To use this, we substitute \( u = 1 \) into the general form of our antiderivative: \[ F(1) = 2 \ln |1| + \frac{1^2}{2} + C = 5 \]Using this equation, we simplify:

• Since \( \ln 1 = 0 \), this term disappears.
• \( \frac{1^2}{2} \) simplifies to \( \frac{1}{2} \).

Thus, the equation becomes \( \frac{1}{2} + C = 5 \), which helps us solve for the constant \( C \). This information is crucial as it allows us to specify the exact antiderivative.

The constant of integration, usually represented by \( C \), plays a pivotal role in calculus when finding antiderivatives. It accounts for the fact that differentiating a constant gives zero, which means when you integrate, there's an unknown constant you should include.In the process of integration, \( C \) ensures that we cover all possible vertical shifts of the function. Since integration itself cannot determine this constant, external information like initial conditions is used to find \( C \) and thus, specify the antiderivative fully.In our problem, once the initial condition \( F(1) = 5 \) was applied, we solved for \( C \) using the simplified equation: \[ \frac{1}{2} + C = 5 \]Therefore, \( C = \frac{9}{2} \).This constant now provides the detail necessary to describe the antiderivative faithfully in accordance with the initial condition. It highlights an essential aspect of indefinite integration, which concerns finding solutions that both cover general cases and meet specific given scenarios.