1)Đặt \(1+2x=t\Leftrightarrow x=\frac{t-1}{2}; dx=\frac{dt}{2}.\)
\(I_1=\frac{1}{4}\int\frac{t-1}{t^3}dt=\frac{1}{4}\int\left(\frac{1}{t^2}-\frac{1}{t^3}\right)dt=...\)
2) \(\int\frac{1-x^2}{x+x^3}dx=\int\left(\frac{1}{x}-\frac{2x}{1+x^2}\right)dx=\int\frac{dx}{x}-\int\frac{d\left(1+x^2\right)}{1+x^2}=...\)
Tìm các nguyên hàm sau
1.\(\int\frac{9x^2}{\sqrt{1-x^3}}dx\)
2.\(\int\frac{1}{\sqrt{x}\left(1+\sqrt{x}\right)^3}dx\)
3.\(\int\frac{x}{\sqrt{2x+3}}dx\)
4.\(\int\) \(\frac{e^{2x}}{\sqrt{1+e^x}}\) dx
5.\(\int\frac{\sqrt[3]{1+lnx}}{x}dx\)
6.\(\int\) cosxsin3xdx
7.\(\int\) (x2+2x-1)exdx
8.\(\int\) excosxdx
9.\(\int\) xsin(2x+1)dx
10.\(\int\) (1-2x)e3xdx
1) \(\int\left(\frac{lnx}{2+lnx}\right)^2\)
2) \(\int\frac{dx}{\left(x+3\right)^3\left(x+5\right)^5}\)
3) \(\int\frac{xdx}{\sqrt{1+\sqrt[3]{x^2}}}\)
4) \(\int\frac{dx}{x^3.\sqrt[3]{2-x^3}}\)
5)\(\int\sqrt[3]{\frac{2-x}{2+x}}.\frac{1}{\left(2-x\right)^2}dx\)
1)\(\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx\)
2)\(\int\frac{dx}{\left(e^x+1\right)\left(x^2+1\right)}\)
3)\(\int\frac{1+2x\sqrt{1-x^2}+2x^2}{1+x+\sqrt{1+x^2}}\)dx
4)\(\int\frac{sin^6x+c\text{os}^6x}{1+6^x}dx\)
5)\(\int_0^{\frac{\pi}{2}}\frac{\sqrt{c\text{os}x}}{\sqrt{s\text{inx}}+\sqrt{c\text{os}x}}dx\)
6)\(\int\frac{x^4}{2^x+1}dx\)
7)\(\int_0^{\frac{\pi^2}{4}}sin\sqrt{x}dx\)
8)\(\int\sqrt[6]{1-c\text{os}^3x}.s\text{inx}.c\text{os}^5xdx\)
9)\(\int\sqrt{\frac{1}{4x}+\frac{\sqrt{x}+e^x}{\sqrt{x}.e^x}}dx\)
10)\(\int\frac{c\text{os}x+s\text{inx}}{\left(e^xs\text{inx}+1\right)s\text{inx}}dx\)
Tìm nguyên hàm các hàm số hữu tỉ sau :
a) \(\int\frac{x^2+2x-1}{\left(x-1\right)\left(x^2+1\right)}dx\)
b) \(\int\frac{x^2+1}{\left(x-1\right)^3\left(x+3\right)}dx\)
Tìm nguyên hàm các hàm số hữu tỉ sau :
a) \(\int\frac{2\left(x+1\right)}{x^2_{ }+2x-3}dx\)
b)\(\int\frac{2\left(x-2\right)dx}{x^2-4x+4}\)
Tính tích phân bất định hàm số hữu tỉ sau :
a) \(\int\frac{dx}{\sqrt{\left(1-x^2\right)^3}}\)
b) \(\int\frac{dx}{\sqrt{x^2+2x+3}}\)
1) \(\int\frac{xdx}{1+\sqrt{x-1}}\)
2) \(\int\frac{sin2xdx}{\cos^3x-\sin^2x-1}\)
3) \(\int\frac{dx}{1+\sqrt{x}+\sqrt{1+x}}\)
4) \(\int\frac{dx}{3x^3+x^2-4x}\)
5) \(\int\frac{dx}{\sqrt{9-x^2}}\)
Tính các nguyên hàm sau :
a) \(\int x\left(3-x\right)^5dx\)
b) \(\int\left(2^x-3^x\right)^2dx\)
c) \(\int x\sqrt{2-5x}dx\)
d) \(\int\dfrac{\ln\left(\cos x\right)}{\cos^2x}dx\)
e) \(\int\dfrac{x}{\sin^2x}dx\)
\(\int\dfrac{x+1}{\left(x-2\right)\left(x+3\right)}dx\)
h) \(\int\dfrac{1}{1-\sqrt{x}}dx\)
i) \(\int\sin3x\cos2xdx\)
k) \(\int\dfrac{\sin^3x}{\cos^2x}dx\)
l) \(\int\dfrac{\sin x\cos x}{\sqrt{a^2\sin^2x+b^2\cos^2x}}dx\) (\(a^2\ne b^2\))
1, \(\int\dfrac{x}{1-cos2x}dx\)
2, \(\int cos2x.e^{3x}dx\)
3, \(\int\left(2x+1\right)ln^2dx\)
4, \(\int\left(2x-1\right)cosxdx\)
5, \(\int\left(x^2+x+1\right)e^xdx\)
6, \(\int\left(2x+1\right)ln\left(x+2\right)dx\)