Nguyên hàm bài tập phần 10

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

Đặt \( t = e^x \Rightarrow dt = e^x dx \Rightarrow dx = \frac{1}{t} \, dt \).

\( \int \frac{1}{e^{2x} - 4e^x + 3} \, dx = \int \frac{1}{t (t^2 - 4t + 3)} \, dt \).

\( \frac{1}{4(t^2 - 4t + 3)} = \frac{1}{t(t-1)(t-3)} = \frac{A}{t} + \frac{B}{t-1} + \frac{C}{t-3} \)

\( \Rightarrow
\begin{cases}
A = \frac{1}{3} \\
B = -\frac{1}{2} \\
C = \frac{3}{2}
\end{cases} \)

\( I = \int \left( \frac{1}{3t} - \frac{1}{2(t-1)} + \frac{3}{2(t-3)} \right) dt \)

\( = \frac{1}{3} \ln|t| - \frac{1}{2} \ln|t-1| + \frac{3}{2} \ln|t-3| + c \)

\( = \frac{1}{3} \ln|e^x| - \frac{1}{2} \ln|e^x - 1| + \frac{3}{2} \ln|e^x - 3| + c. \)

page50


\( \int \frac{2^x - 1}{e^x} \, dx \quad (SGK chuẩn) \)

Nhắc: \( \int a^x \, dx = \frac{a^x}{\ln a} + c \)

\( \int \frac{2^x - 1}{e^x} \, dx = \int \left[ \left( \frac{2}{e} \right)^x - \left( \frac{1}{e} \right)^x \right] dx \)

\( = \frac{\left( \frac{2}{e} \right)^x}{\ln \left( \frac{2}{e} \right)} - \frac{\left( \frac{1}{e} \right)^x}{\ln \left( \frac{1}{e} \right)} + c \)

\( = \frac{\left( \frac{2}{e} \right)^x}{\ln 2 - 1} - \frac{\left( \frac{1}{e} \right)^x}{-1} + c \)

\( = \frac{1}{\ln 2 - 1} \cdot \left( \frac{2}{e} \right)^x + \frac{1}{e^x} + c. \)

page51


\( \int \frac{\cos x}{1 + \cos x} \, dx \)

\( \int \frac{\cos x}{1 + \cos x} \, dx = \int \frac{\cos x + 1 - 1}{1 + \cos x} \, dx = \int dx - \int \frac{1}{1 + \cos x} \, dx \)

\( = x - \int \frac{1}{2 \cos^2 \frac{x}{2}} \, dx = x - 2 \tan \frac{x}{2} + c \)

page52


\( \int \frac{\cos x - \sin x}{\sin x + \cos x} \, dx \)

\( \int \frac{\cos x - \sin x}{\sin x + \cos x} \, dx = \int \frac{u'}{u} \, dx \quad \text{với } u = \sin x + \cos x \)

\( = \ln |\sin x + \cos x| + C \)

page53


\( \int \frac{\cos x}{\sin x + \cos x} \, dx = I \)

Xét \( \int \frac{\sin x}{\sin x + \cos x} \, dx = J \)

\( I + J = \int \frac{\cos x}{\sin x + \cos x} \, dx + \int \frac{\sin x}{\sin x + \cos x} \, dx = x + C \)

\( I - J = \int \frac{\cos x}{\sin x + \cos x} \, dx - \int \frac{\sin x}{\sin x + \cos x} \, dx = \ln |\sin x + \cos x| + C \)

\( I = \int \frac{\cos x}{\sin x + \cos x} \, dx = \frac{1}{2} \left( x + \ln |\sin x + \cos x| \right) + C. \)

page54