\(u=3x+1\Rightarrow x=\dfrac{u-1}{3}\Rightarrow du=3dx\)
\(\Rightarrow\int x\left(3x+1\right)^9dx=\int\dfrac{1}{3}\left(u-1\right).u^9.\dfrac{du}{3}=\dfrac{1}{9}\int\left(u^{10}-u^9\right)du\)
\(=\dfrac{1}{9}\int u^{10}du-\dfrac{1}{9}\int u^9du=\dfrac{1}{99}.u^{11}-\dfrac{1}{90}.u^{10}=\dfrac{1}{99}\left(3x+1\right)^{11}-\dfrac{1}{90}.\left(3x+1\right)^{10}\)
Sử dụng phương pháp biến đổi số, hãy tính :
a) \(\int\left(1-x\right)^9dx\) (đặt \(u=1-x\))
b) \(\int x\left(1+x^2\right)^{\dfrac{2}{3}}dx\) (đặt \(u=1+x^2\))
c) \(\int\cos^3x\sin x.dx\) (đặt \(t=\cos x\))
d) \(\int\dfrac{dx}{e^x+e^{-x}+2}\) (đặt \(u=e^x+1\))
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, \(\int\left(2x+1\right)ln\left(x+2\right)dx\)
Tìm nguyên làm các hàm số hữu tỉ sau :
a)
\(\int\frac{3x^2+3x+12}{\left(x-1\right)\left(x+2\right)}dx\)
b) \(\int\frac{x^2+2x+6}{\left(x-1\right)\left(x-2\right)\left(x-4\right)}dx\)
\(\int\dfrac{1}{cosx.cos\left(x+\dfrac{\pi}{4}\right)}dx\)
\(\int\dfrac{1}{x^3\left(1+x^2\right)}dx=\dfrac{a}{x^2}+blnx+cln\left(1+x^2\right).S=a+b+c=?\)
\(\int\dfrac{5-3x}{\left(x^2-5x+6\right)\left(x^2-2x+1\right)}dx=\dfrac{a}{x-1}-ln\left(\dfrac{x-b}{x-c}\right)+C.P=2a+b\)
\(\int\dfrac{3x+1}{\left(x+1\right)^3}dx\)
Tìm nguyên hàm:
a) \(\int\left(\dfrac{1}{u^3}+\dfrac{1}{u^2}+\dfrac{1}{u}\right)du\)
b) \(\int\left(\dfrac{1}{t-2}+\dfrac{3}{1-t}\right)dt\)
c) \(\int\left(\dfrac{1}{2-3x}+\dfrac{7}{1-4x}\right)dx\)
d) \(\int e^{5x-1}dx\)
\(\int\frac{tan^3x}{c\text{os}2x}dx\)
2) \(\int\frac{xe^x\left(4+4\left(s\text{inx}+c\text{os}x\right)+sin2x\right)}{\left(1+c\text{os}x\right)^2}\)
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) \(\int\dfrac{\sin x\cos x}{\sqrt{a^2\sin^2x+b^2\cos^2x}}dx\) (\(a^2\ne b^2\))
Tìm: I = \(\int x^2\left(2-3x^2\right)^8dx\)
