\(u=x+3\Rightarrow x=u-3\Rightarrow du=dx\)
\(\Rightarrow\int\left(2x+1\right)\left(x+3\right)^4dx=\int\left[2\left(u-3\right)+1\right].u^4du\)
\(=\int\left(2u-5\right)u^4du=\int(2u^5-5u^4)du=\dfrac{2}{6}u^6-\dfrac{5}{5}u^5=\dfrac{1}{3}u^6-u^5=\dfrac{1}{3}\left(x+3\right)^6-\left(x+3\right)^5\)
1, \(\int sin2x.e^{3x}dx\)
2, \(\int\dfrac{x^4dx}{\left(x^2-1\right)^2}\)
3, \(\int e^x.cos^2xdx\)
4, \(\int e^{2x}sin^2xdx\)
5, \(\int e^{-x}.cos^3xdx\)
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 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) \(\int\frac{x^2+1}{\left(x-1\right)^3\left(x+3\right)}dx\)
\(\int\dfrac{1}{cosx.cos\left(x+\dfrac{\pi}{4}\right)}dx\)
\(\int\dfrac{1}{x^3\left(1+x^2\right)}dx=\dfrac{a}{x^2}+blnx+cln\left(1+x^2\right).S=a+b+c=?\)
\(\int\dfrac{5-3x}{\left(x^2-5x+6\right)\left(x^2-2x+1\right)}dx=\dfrac{a}{x-1}-ln\left(\dfrac{x-b}{x-c}\right)+C.P=2a+b\)
1, \(\int\dfrac{lnxdx}{\sqrt{x}}\)
2, \(\int ln\left(x+\sqrt{x^2+1}\right)dx\)
3, \(\int\left(x^2+2x+3\right)dx\)
Sử dụng phương pháp tính nguyên hàm từng phần, hãy tính :
a) \(\int x\ln\left(1+x\right)dx\)
b) \(\int\left(x^2+2x-1\right)e^xdx\)
c) \(\int x\sin\left(2x+1\right)dx\)
d) \(\int\left(1-x\right)\cos xdx\)
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) \(\int\dfrac{\sin x\cos x}{\sqrt{a^2\sin^2x+b^2\cos^2x}}dx\) (\(a^2\ne b^2\))
\(\int\frac{x}{\left(1+2x\right)^3}dx\)
\(\int\frac{1-x^2}{x+x^3}dx\)
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) \(\int\frac{x^2+2x+6}{\left(x-1\right)\left(x-2\right)\left(x-4\right)}dx\)
