Math Problem Statement

sin(x-1)的积分表达式

Solution

对函数 sin(x1)\sin(x-1) 进行不定积分,按照积分规则来求解:

sin(x1)dx\int \sin(x - 1) \, dx

我们可以通过变量替换法进行解题。令 u=x1u = x - 1,则有:

du=dxdu = dx

因此,积分可以重写为:

sin(u)du=cos(u)+C\int \sin(u) \, du = -\cos(u) + C

再将 u=x1u = x - 1 代回:

cos(x1)+C-\cos(x - 1) + C

因此,sin(x1)dx\int \sin(x - 1) \, dx 的不定积分表达式为:

cos(x1)+C-\cos(x - 1) + C

相关问题:

  1. 如何对 cos(x1)\cos(x - 1) 进行不定积分?
  2. 如何对三角函数的乘积进行积分?
  3. 如果已知定积分区间,如何求 sin(x1)\sin(x-1) 的定积分?
  4. 如何使用分部积分法处理三角函数?
  5. 如何求 sin2(x1)\sin^2(x-1) 的积分?

提示:

积分中变量替换法是一种非常常见且实用的技巧,适用于许多类型的积分问题。

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Trigonometric Functions

Formulas

∫sin(u) du = -cos(u) + C
u-substitution: u = x - 1

Theorems

Fundamental Theorem of Calculus
Integration by Substitution

Suitable Grade Level

Grades 11-12, College Level Calculus