Determine the equation of a curve in the xy-plane that passes through the
point (0, 1) and has the slope x2 sin 4x at any point (x, y) on the curve.
To determine the equation of the curve, we can start by integrating the given slope expression to obtain the equation for y in terms of x.
The slope of the curve at any point (x, y) is given by the expression: dy/dx = x^2 * sin(4x)
To find y, we need to integrate this expression with respect to x.
∫ dy = ∫ x^2 * sin(4x) dx
Integrating both sides:
y = ∫ x^2 * sin(4x) dx
To evaluate this integral, we will use integration by parts:
Let u = x^2 and dv = sin(4x) dx. Then, du = 2x dx and v = -1/4 cos(4x).
Using the formula for integration by parts:
∫ u dv = uv - ∫ v du
∫ x^2 * sin(4x) dx = -1/4x^2 * cos(4x) + ∫ 1/2x * cos(4x) dx
Now, we need to evaluate the second integral on the right-hand side. Again, we can apply integration by parts:
Let u = 1/2x and dv = cos(4x) dx. Then, du = -1/2x^2 dx and v = 1/4 sin(4x).
Using the integration by parts formula:
∫ u dv = uv - ∫ v du
∫ 1/2x * cos(4x) dx = 1/4x * sin(4x) + ∫ 1/4 * sin(4x) / x^2 dx
The integral on the right-hand side is known as the sine integral Si(x). Therefore:
∫ 1/4 * sin(4x) / x^2 dx = 1/4 * Si(4x)/4
Plugging this result back into the previous expression:
∫ x^2 * sin(4x) dx = -1/4x^2 * cos(4x) + 1/4x * sin(4x) + 1/4 * Si(4x)/4
Adding the constant of integration, C, to the right-hand side, we obtain the equation for y in terms of x:
y = -1/4x^2 * cos(4x) + 1/4x * sin(4x) + 1/4 * Si(4x)/4 + C
To find the specific equation of the curve passing through the point (0, 1), we can substitute these coordinates into the equation:
1 = -1/4(0)^2 * cos(4(0)) + 1/4(0) * sin(4(0)) + 1/4 * Si(4(0))/4 + C
Simplifying this expression:
1 = 0 + 0 + 0 + C
1 = C
The constant of integration C is equal to 1. Substituting this value back into the equation for y, we obtain the final equation of the curve:
y = -1/4x^2 * cos(4x) + 1/4x * sin(4x) + 1/4 * Si(4x)/4 + 1
Therefore, the equation of the curve in the xy-plane that passes through the point (0, 1) and has the slope x^2 * sin(4x) at any point (x, y) on the curve is y = -1/4x^2 * cos(4x) + 1/4x * sin(4x) + 1/4 * Si(4x)/4 + 1.
To determine the equation of the curve that passes through the point (0, 1) and has the given slope, we can use the method of integration.
Step 1: Find the equation of the slope function.
Given that the slope at any point (x, y) on the curve is given by dy/dx = x^2 * sin(4x), we integrate this expression to find the equation of the slope function. Let's denote the slope function as f(x):
∫ (x^2 * sin(4x)) dx = f(x)
Step 2: Solve for f(x).
Integrating x^2 * sin(4x) can be done by applying integration by parts. The formula for integrating u dv is ∫ u dv = uv - ∫ v du.
Let's assign u as x^2 and dv as sin(4x) dx. Then, differentiate u to get du = 2x dx, and integrate dv to get v = -1/4 cos(4x).
Applying the integration by parts formula:
∫ (x^2 * sin(4x)) dx = -1/4 x^2 cos(4x) - ∫ (-1/4 cos(4x) * 2x) dx
Simplifying further:
∫ (x^2 * sin(4x)) dx = -1/4 x^2 cos(4x) + 1/2 ∫ (x * cos(4x)) dx
The integral of x * cos(4x) can be solved by applying integration by parts again. Let's assign u as x and dv as cos(4x) dx. Then, differentiate u to get du = dx and integrate dv to get v = 1/4 sin(4x).
Applying the integration by parts formula again:
∫ (x * cos(4x)) dx = (1/4 x sin(4x)) - ∫ (1/4 sin(4x) dx)
Simplifying further:
∫ (x * cos(4x)) dx = (1/4 x sin(4x)) - 1/16 cos(4x)
Substituting back into the previous expression:
∫ (x^2 * sin(4x)) dx = -1/4 x^2 cos(4x) + 1/2 ((1/4 x sin(4x)) - 1/16 cos(4x))
This gives the equation for the slope function f(x):
f(x) = -1/4 x^2 cos(4x) + 1/2 ((1/4 x sin(4x)) - 1/16 cos(4x))
Step 3: Find the equation of the curve.
We have the slope function f(x), and we know that the curve passes through the point (0, 1). We can use this information to find the equation of the curve.
To find the curve, we need to integrate the slope function f(x) to get the equation for the curve. Let's denote the equation of the curve as y = g(x):
∫ f(x) dx = ∫ (dy/dx) dx = ∫ dx/dy dy = ∫ dy = g(x) + C
To find the value of C, we substitute the coordinates of the point (0, 1) into the equation of the curve:
1 = g(0) + C
Since the slope at any point (x, y) on the curve is given by f(x), the equation of the curve is given by:
y = g(x) + C
where g(x) is the antiderivative of f(x), and C is a constant determined by the point (0, 1).