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Lý thuyết Nguyên hàm Toán 12 Cánh Diều

1. Khái niệm nguyên hàm Cho hàm số f(x) xác định trên K. Hàm số F(x) được gọi là một nguyên hàm của hàm số f(x) trên K nếu F’(x)=f(x) với mọi x thuộc K.

ܫTổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Cánh diều

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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1. Khái niệm nguyên hàm

Cho hàm số f(x) xác định trên K. Hàm số F(x) được gọi là một nguyên hàm của hàm số f(x) trên K nếu F’(x)=f(x) với mọi x thuộc K.

Chú ý:

Cho K là một khoảng, đoạn hoặc nửa khoảng của tập số thực RGiả sử hàm số F(x) là một nguyên hàm của f(x) trên K. Khi đó:a) Với mỗi hằng số C, hàm số F(x) + C cũng là một nguyên hàm của f(x) trên Kb) Nếu hàm số G(x) là một nguyên hàm của f(x) trên K thì tồn tại một hằng số C sao chp G(x) = F(x) + C với mọi x thuộc KHọ (hay tập hợp) tất cả các nguyên hàm của hàm số f(x) trên K được kí hiệu là\(\int {f(x)dx = F(x) + C} \)

2. Tính chất của nguyên hàm

  • \(\int {kf(x)dx = k\int {f(x)dx(k \ne 0)} } \)
  • \(\int {\left[ {f(x) + g(x)} \right]} dx = \int {f(x)dx + \int {g(x)dx} } \)
  • \(\int {\left[ {f(x) - g(x)} \right]} dx = \int {f(x)dx - \int {g(x)dx} } \)
 

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