Giải bài tập 4.10 trang 19 SGK Toán 12 tập 2 - Cùng khám phá

Biết (F(x) = sqrt x ) là một nguyên hàm của hàm số (f(x)). Tính (int_1^4 {left[ {2 + f(x)} right]dx} ).

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Đề bài

Biết \(F(x) = \sqrt x \) là một nguyên hàm của hàm số \(f(x)\). Tính \(\int_1^4 {\left[ {2 + f(x)} \right]dx} \).

Phương pháp giải - Xem chi tiết

Sử dụng định nghĩa của nguyên hàm: \(F'(x) = f(x)\). Áp dụng tính chất của tích phân: \(\int {(a + b)dx = \int {adx} + \int {bdx} } \) Tính riêng tích phân của các hàm hằng và hàm \(f(x)\).

Lời giải chi tiết

\(\int_1^4 {\left[ {2 + f(x)} \right]dx} = \int_1^4 2 dx + \int_1^4 f (x)dx\) Tích phân thứ nhất: \(\int_1^4 2 dx = 2x|_1^4 = 2(4) - 2(1) = 8 - 2 = 6\) Tích phân thứ hai: \(\int_1^4 f (x)dx = F(x)|_1^4 = \sqrt 4 - \sqrt 1 = 2 - 1 = 1\) Vậy: \(\int_1^4 {\left[ {2 + f(x)} \right]dx} = 6 + 1 = 7\)

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4.9 trên 7 phiếu

💖 Giải bài tập 4.11 trang 19 SGK Toán 12 tập 2 - Cùng khám phá Cho hàm số \(f(x)\) có đạo hàm \(f'(x)\) liên tục trên đoạn \([ - 1;4]\) thỏa mãn \(f( - 1) = 2\), \(f(4) = 7\). Tính \(\int_{ - 1}^4 {f'} (x)dx\).
𓆉 Giải bài tập 4.12 trang 19 SGK Toán 12 tập 2 - Cùng khám phá Cho các hàm số \(f(x)\), \(g(x)\) liên tục trên đoạn \([ - 1;3]\) thỏa mãn \(\int_{ - 1}^2 f (x)dx = 2\), \(\int_{ - 1}^3 f (x)dx = 6\), và \(\int_{ - 1}^2 g (x)dx = - 1\). Tính: a) \(\int_2^3 f (x)dx\); b) \(I = \int_{ - 1}^2 {\left( {x + 2f(x) - 3g(x)} \right)} dx\).
🅘 Giải bài tập 4.13 trang 20 SGK Toán 12 tập 2 - Cùng khám phá Tính các tích phân sau: a) \(\int_{ - 1}^2 x (x + 1)dx\); b) \(\int_0^{\frac{\pi }{2}} {{{\cos }^2}} \frac{x}{2}dx\); c) \(\int_1^2 {{2^{1 - 3x}}} dx\); d) \(\int_0^{\frac{\pi }{4}} {{{\tan }^2}} xdx\); e) \(\int_1^4 {\left( {{e^{2x + 1}} - 3x\sqrt x } \right)} dx\); g) \(\int_1^4 | 5 - 3x|dx\).
🙈 Giải bài tập 4.14 trang 20 SGK Toán 12 tập 2 - Cùng khám phá Một quả bóng được ném lên từ độ cao \(1,5m\) với vận tốc ban đầu \(24m/s\). Biết gia tốc của quả bóng là \(a = - 9,8m/{s^2}\). a) Tính vận tốc của quả bóng tại thời điểm 1 giây sau khi được ném lên. b) Tính quãng đường quả bóng đi được từ lúc ném lên đến khi chạm đất lần đầu.
♎ Giải bài tập 4.15 trang 20 SGK Toán 12 tập 2 - Cùng khám phá
Đường gấp khúc ABD trong Hình 4.8 là đồ thị vận tốc \(v(t)\) của một vật (t = 0 là thời điểm vật bắt đầu chuyển động). Trong khoảng thời gian mà \(v < 0\)thì vật chuyển động ngược chiều với khoảng thời gian mà \(v >✤ 0\). a) Viết công thức của hàm số \(v(t)\) với \(t \in [0;9]\). b) Biết rằng quãng đường vật đi chuyển với vận tốc \(v = v(t)\) từ thời điểm \(t = a\) đến thời điểm \(t = b\) là \(s = \int_a^b | v(t)|{\mkern 1mu} dt\), tính quãng đường vật di chuyển được trong 9 giây kể từ khi vật

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