Lời giải:
\(\int \frac{dx}{e^x+1}=\int \frac{e^xdx}{e^x(e^x+1)}=\int \frac{d(e^x)}{e^x(e^x+1)}=\int \left ( \frac{d(e^x)}{e^x}-\frac{d(e^x+1)}{e^x+1} \right )\)
\(=\ln|e^x|-ln|e^x+1|+c=x-\ln(e^x+1)+c\)
Tính nguyên hàm của:
1, \(\int\)\(\dfrac{x^3}{x-2}dx\)
2, \(\int\)\(\dfrac{dx}{x\sqrt{x^2+1}}\)
3, \(\int\)\((\dfrac{5}{x}+\sqrt{x^3})dx\)
4, \(\int\)\(\dfrac{x\sqrt{x}+\sqrt{x}}{x^2}dx\)
5, \(\int\)\(\dfrac{dx}{\sqrt{1-x^2}}\)
\(\int\limits^{ln3}_{ln2}\frac{1}{e^x-1}dx\)
a) \(\int sin^2\frac{x}{2}dx\)
b) \(\int cos^2\frac{x}{2}dx\)
c) \(\int\frac{2x+1}{x^2+x+5}dx\)
d) \(\int\left(2tanx+cotx\right)^2dx\)
cho \(\int\limits^2_0\frac{dx}{x^2-x+1}=\int\limits^{\frac{\pi}{3}}_{-\frac{\pi}{6}}\frac{2}{a}dx\) . Chon khẳng định đúng
Tính tích phân :
\(\int^1_0\left(\frac{x^2-4x+3}{e^{2x}}\right)dx\)
Tính nguyên hàm \(\int e^x\left(2-x\right)dx\)
Tìm các nguyên hàm sau đây
a) \(I_1=\int e^{2x}\sin3xdx\)
b) \(I_2=\int e^{-x}\cos\frac{x}{2}dx\)
c) \(I_2=\int e^{3x}\cos\left(e^x\right)d\)
Tính tích phân: \(\int\limits^{log\left(1+\sqrt{2}\right)}_0\left(\dfrac{e^x-e^{-x}}{2}\right)^3\cdot\left(\dfrac{e^x+e^{-x}}{2}\right)^{11}dx\)
Tính tích phân : \(I=\int\limits^2_0\frac{x^5}{\sqrt{x^3+1}}dx\)