Let f be a non-negative function defined on [\(0,\frac{\pi}{2}\)]. If \(\int_{0}^{x}(f'(t)-sin\,2t)dt=\int_{0}^{x}f(t)tan\,t\,dt\). \(f(0)=1\), then \(\int_{0}^{\frac{\pi}{2}}f(x)dx\) is
- 3
- \(3-\frac{\pi}{2}\)
- \(3+\frac{\pi}{2}\)
- \(\frac{\pi}{2}\)
The Correct Option is B
Solution and Explanation
The integral equality states that the integral of the derivative of f(x) minus sin2(x) over the interval [0,x] equals the integral of f(x) times tan(x) over the same interval. Given that 1f(0)=1, the value of∫02πf(x)dx is 3−π2. This answer is accurate due to consistent application of integration properties and the initial condition 1f(0)=1.
The correct answer is option (B): \(3-\frac{\pi}{2}\)
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations