Nguyên hàm bài tập phần 2

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Nguyên hàm bài tập phần 2

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. \( F(x) = \frac{2}{3}\sqrt{1+x^2} + \frac{8}{9} \)

\( F(x) = \int x\sqrt{1+x^2} \, dx = \frac{1}{2} \int (2x(1+x^2)^{\frac{1}{2}}) \, dx \)

\( = \frac{1}{2} \frac{(1+x^2)^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{1}{3} (1+x^2)\sqrt{1+x^2} + C \)

\( F(\sqrt{3}) = \frac{8}{3} + C = 2 \implies C = -\frac{2}{3} \implies \boxed{A}\)

Thêm:

\( \int \cos x \sqrt{1+2\sin x} \, dx \)
\( \int x^3 (1+x^4)^5 \, dx \)

page10

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) \(\int \frac{x^2 - 3x + 4}{x-1} \, dx\)

*Tính \(f(3)\), biết \(f'(x) = \frac{2}{x^2+1}\). \(f(1) = \ln 2\):

A. \(f(3) = \ln 20 \quad \) B. \(f(3) = \frac{1}{2} \ln 20\)
C. \(f(3) = \frac{1}{2} \ln 5 \quad\) D. \(f(3) = \frac{1}{2} \ln 10\)

\( f(x) = \int \frac{x}{x^2+1} \, dx = \frac{1}{2} \int \frac{2x}{x^2+1} \, dx = \frac{1}{2} \ln |x^2+1| + C \)

\( f(1) = \frac{1}{2} \ln 2 + C = \ln 2 \implies C = \frac{1}{2} \ln 2 \)

\( f(x) = \frac{1}{2} \ln |x^2+1| + \frac{1}{2} \ln 2 \implies f(3) = \frac{1}{2} \ln 10 + \frac{1}{2} \ln 2 = \frac{1}{2} \ln 20 \implies \boxed{B}\)

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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) \(\int \frac{2x+3}{x^2} \, dx\)

page12

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)

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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) \(\int \cos x \, dx = \sin x + C, \quad \int \cos u \, du = \sin u + C.\)
\(\int \sin x \, dx = -\cos x + C, \quad \int \sin u \, du = -\cos u + C.\)

* \(\int u' \cos u \, dx = \int \cos u \, du = \sin u + C\)
\(\int \cos ax \, dx = \frac{1}{a} \sin ax + C\)

* \(\int u' \sin u \, dx = \int \sin u \, du = -\cos u + C\)
\(\int \sin ax \, dx = -\frac{1}{a} \cos ax + C\)

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