Bài tập: Nguyên hàm \( F(x)\) của hàm số \( f(x) = x\sqrt{1+x^2} \) thỏa mãn điều kiện \( F(\sqrt{3}) = 2 \) là:
A. \( F(x) = \frac{1}{3}(\sqrt{1+x^2})^3 - \frac{2}{3} \quad\)
B. \( F(x) = \frac{2}{3}\sqrt{1+x^2} + \frac{2}{3} \)
C. \( F(x) = \frac{2}{3}(\sqrt{1+x^2})^3 - \frac{2}{3} \quad\)
D. \( F(x) = \frac{2}{3}(\sqrt{1+x^2})^3 + \frac{8}{9} \)
Đáp án
page 10
3) • \(\int \frac{1}{x} \, dx = \ln |x| + C\), \(\int \frac{1}{u} \, du = \ln |u| + C\)
• \(\int \frac{u'}{u} \, dx = \int \frac{1}{u} \, du = \ln |u| + C\)
a) \(\int \frac{1}{3x-2} \, dx\)
• \( \int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln |ax+b| + C \)
b) \(\int \frac{4x-3}{x-2} \, dx\)
c) \(\int \frac{x+2}{2x-1} \, dx = \int \frac{\frac{1}{2}(2x-1)+\frac{5}{2}}{2x-1} \, dx\)
d) \(\int \frac{x^2 - 3x + 4}{x-1} \, dx\)
Bài tập: Tính \(f(3)\), biết \(f'(x) = \frac{x}{x^2+1}\) và \(f(1) = \ln 2\)
A. \(f(3) = \ln 20 \quad \)
B. \(f(3) = \frac{1}{2} \ln 20\)
C. \(f(3) = \frac{1}{2} \ln 5 \quad\)
D. \(f(3) = \frac{1}{2} \ln 10\)
Đáp án
page 11
f) \(\int \tan x \, dx\)
g) \(\int \tan^3 x \, dx. \quad \int \tan^2 x \, dx \)
h) \(\int \frac{1}{\sin x \cos x} \, dx = \int \frac{\frac{1}{\cos^2 x}}{\tan x} \, dx = \ln |\tan x| + C\)
• \(\int \frac{1}{\sin 2x} \, dx\)
• \(\int \frac{1}{\sin x} \, dx = \frac{1}{2} \int \frac{\frac{1}{\cos^2 \frac{x}{2}}}{ \tan \frac{x}{2}} \, dx = \ln |\tan \frac{x}{2}| + C\)
i) \(\int \frac{1}{x \ln x} \, dx = \ln |\ln x| + C\)
k) \(\int \frac{\cos x}{3\sin x - 4} \, dx\)
e) \(\int \frac{2x+3}{x^2} \, dx\)
page 12
4) \(\int e^x \, dx = e^x + C, \quad \int e^u \, du = e^u + C\)
\(\int u' e^u \, dx = \int e^u \, du = e^u + C\)
a) \(\int e^{3x+1} \, dx\)
b) \(\int \cos x \, e^{\sin{x}} \, dx\)
c) \(\int \frac{e^x}{1+e^x} \, dx\)
d) \(\int \frac{1}{1+e^x} \, dx\) (thêm bớt)
page 13
5) \(a > 0, a \neq 1\):
\(\int a^x \, dx = \frac{a^x}{\ln a} + C, \quad \int a^u \, du = \frac{a^u}{\ln a} + C\)
Ví dụ:
\(\int (x^3 + 3^x) \, dx\)
6) \(\int \cos x \, dx = \sin x + C, \quad \int \cos u \, du = \sin u + C\)
\(\int \sin x \, dx = -\cos x + C, \quad \int \sin u \, du = -\cos u + C\)
• \(\int u' \cos u \, dx = \int \cos u \, du = \sin u + C\)
• \(\int \cos ax \, dx = \frac{1}{a} \sin ax + C\)
• \(\int u' \sin u \, dx = \int \sin u \, du = -\cos u + C\)
• \(\int \sin ax \, dx = -\frac{1}{a} \cos ax + C\)
page 14