Question

In Exercises \(21-26,\) construct a function of the form \(y=\int^{x} f(t) d t+C\) that satisfies the given conditions. $$\frac{d y}{d x}=\ln (\sin x+5),\( and \)y=3\( when \)x=2$$

Short answer

The answer will be the complete function \( y \) that comes from step 2, where we have found the constant \( C \) by using the given point, and have applied the integral to find the function from its derivative.

Step-by-step solution

Integrate the Function

The function \( \frac{d y}{d x}=\ln (\sin x+5) \) can be integrated using the basic rules of integral calculus. The integral of \( \ln (\sin x+5) \) with respect to \( x \) will provide the original function minus the constant of integration, \( C \). This gives us \( y = \int \ln (\sin x+5) dx \).

Finding the Constant of Integration

For the initial condition \( y=3 \) when \( x=2 \), we substitute \( x=2 \), \( y=3 \) into our function and solve for \( C \). Therefore, the function would be \( 3= \int \ln (\sin 2+5) dx + C \). Once we evaluate the integral, we solve for \( C \). This gives the complete function.

Key concepts

Integration

If you've ever wondered about the reverse process of taking a derivative, then you've ventured into the realm of integration. Integration is a fundamental operation in calculus that essentially asks the question: 'Given a rate of change, what is the function that represents the accumulated change?' Imagine we have a function that describes the rate at which a car's speed is changing. Integration would help us to find the function that describes the car's total distance traveled.

Let's say you've been given the derivative of a function, such as \( \frac{d y}{d x}=\ln (\sin x+5) \). To unravel this, which means to find the original function whose derivative is \( \ln (\sin x+5) \), you need to integrate. In our example, the integration of \( \ln (\sin x+5) \) with respect to \( x \) is performed to get the function \( y \) — a process akin to piecing together a fragmented story to see the full narrative. It's like saying, 'If this is the rate of change, what was the amount that changed?' Integration provides us with a powerful tool for understanding the cumulative effect of change.

Constant of Integration

When integrating a function, we're often faced with an interesting quirk: there are actually infinite functions that could have produced the same derivative. This is where the constant of integration, usually denoted as \( C \), comes to play. It's a bit like being told a story from the middle and realizing there are many possible beginning chapters that could have led to this point.

Imagine you put together a jigsaw puzzle, but you find there's one piece that could fit in many places - that's our \( C \). To illustrate, when we integrate \( \ln (\sin x+5) \), we say \( y = \int \ln (\sin x+5) dx + C \), acknowledging we're not just finding a specific function, but a whole family of functions that differ only by a constant value. This little \( C \) is vital because it allows us to tailor the function to specific scenarios by setting initial conditions or other known values.

Initial Condition

In the grand tapestry of calculus, the initial condition plays the role of a tailor, fine-tuning the general solution of an integral to fit the precise measurements of our problem. The initial condition provides a specific snapshot in time — a moment where we know exactly where to place our story's protagonist on their journey.

Let's consider the equation \( y=3 \) when \( x=2 \). This knowledge is like having a photograph of a moment along our function's journey — we know exactly where we are at that instant. We use this 'photograph' to find the exact value of our mysterious constant \( C \) in the equation \( y = \int \ln (\sin x+5) dx + C \). By plugging in the initial conditions into our integrated function, we compare it to the 'photograph', which tells us exactly what \( C \) needs to be. Thus, the initial condition helps us transition from the realm of the general to the specific, making our integral solution not just a possibility, but the reality of the situation.