How to derive integral(0-1)x^4sin(x-cos(x^2)dy/dt))
To derive the integral ∫(0-1) x^4sin(x-cos(x^2)) (dy/dt) with respect to t, you can follow these steps:
1. Differentiate the integral with respect to t, treating x as a constant. This will allow us to apply the chain rule to the integral.
d/dt ∫(0-1) x^4 sin(x - cos(x^2)) (dy/dt)
2. Apply the chain rule to the integral. We treat x as a variable and differentiate both the integrand and the limits of integration in respect to t.
∫(0-1) [d/dx (x^4 sin(x - cos(x^2))) * dx/dt] dy/dt
3. Evaluate the derivative of the integrand with respect to x.
The derivative of x^4 sin(x - cos(x^2)) with respect to x can be found by applying the product rule:
d/dx (x^4 sin(x - cos(x^2))) = 4x^3 sin(x - cos(x^2)) + x^4 cos(x - cos(x^2))(1 + 2x sin(x^2))
4. Substitute the derivative back into the integral equation.
∫(0-1) [(4x^3 sin(x - cos(x^2)) + x^4 cos(x - cos(x^2))(1 + 2x sin(x^2))) * dx/dt] dy/dt
5. Simplify the integral if necessary and evaluate it based on the given limits.
Please note that the simplification and evaluation of the integral is dependent on the specific values of x and the differential equation dy/dt.
It is worth mentioning that in order to obtain the final expression for the derivative, you may need to use additional differentiation techniques such as product rule, chain rule, or trigonometric identities.