托马斯微积分课件6.3 Derivatives of inverse trigonometric functions and integrals.ppt

* * * * 目录 上页 下页 返回 结束 Chapter 6 Transcendental Functions and Differential Equations 6.3 Derivatives of Inverse Trigonometric Functions; Integrals 6.1 Logarithms 6.2 Exponential Functions 6.4 First-Order Separable Differential Equations 6.5 Linear First-Order Differential Equations 6.6 Euler’s Method; Population Models 6.7 Hyperbolic Functions 6.3 Derivatives of Inverse Trigonometric Functions; Integrals (反三角函数的导数与积分) 6.3.1 Derivatives of the Arcsine 6.3.2 Derivatives of the Arctangent 6.3.3 Derivatives of the Other Two 6.3.4 Integration Formulas 6.3.1 Derivatives of the Arcsine (反正弦函数求导) Consider the function defined by Its inverse function ,denoted by Let us find the derivative. If u is a differentiable function of x with |u|1, we apply the Chain rule to get 6.3.2 Derivatives of the Arctangent (反正切函数求导) Consider the function defined by Its inverse function ,denoted by Let us find the derivative. If u is a differentiable function of x, we apply the Chain rule to get 6.3.3 Derivatives of the Other Two (其他两个反三角函数的导数) Consider the function defined by Its inverse function ,denoted by Let us find the derivative. If u is a differentiable function of x, we apply the Chain rule to get Note that We have that is to say According to the Lagrange Mean Value Theorem furthermore, it is easy to deduce In fact, Thus, the derivative of the Arccotangent To sum up, we give the following table Example 1. Find the derivative Solution. 6.3.4 Integration Formula (积分公式) 1. Solution. 2. Solution. Example 2. Evaluate the definite integral Solution. Example 3-1. Evaluate the indefinite integral Solution. Example 3-2. Evaluate the indefinite integral Solution. Example 3-3. Evaluate the indefinite integral Solution. Example 4-1. Evaluate the indefinite integral Solution. Example 4-2. Evaluate the indefinite integral Solution. Example 5. Solve the Initial Value Problem (IVP) Solution. Integrate both sides , we have therefor