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Lý thuyết Nguyên hàm Toán 12 Chân trời sáng tạo

1. Khái niệm nguyên hàm

♒Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

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:

Giả 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 KNhư vậy, nếu F(x) là một nguyên hàm của f(x) trên K thì mọi nguyên hàm của f(x) trên K đều có dạng F(x) + C (C là hằng số). Ta gọi F(x) + C là họ các nguyên hàm của f(x) trên K, kí hiệu bởi \(\int {f(x)dx} \)

2. Nguyên hàm của một số hàm sơ cấp

a) Nguyên hàm của hàm số lũy thừa

+ \(\int {0dx = C} \) + \(\int {1dx = x + C} \) + \(\int {{x^\alpha }dx = \frac{{{x^{\alpha  + 1}}}}{{\alpha  + 1}} + C(\alpha  \ne  - 1)} \)

b) Nguyên hàm của hàm số \(y = \frac{1}{x}\)

\(\int {\frac{1}{x}x = \ln \left| x \right| + C} \)

c) Nguyên hàm của một số hàm số lượng giác

+ \(\int {\cos xdx = \sin x + C} \) + (\int {\sin xdx =  - \cos x + C} \) + \(\int {\frac{1}{{{{\cos }^2}x}}dx = \tan x + C} \) + \(\int {\frac{1}{{{{\sin }^2}x}}dx =  - \cot x + C} \)

d) Nguyên hàm của hàm số mũ

+ \(\int {{e^x}dx = {e^x} + C} \) + \(\int {{a^x}dx = \frac{{{a^x}}}{{\ln a}} + C(0 < a \ne 1)} \)

3. Tính chất cơ bản 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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