Đặt t=x+2 \(\Rightarrow\) dt=dx.
\(\int\dfrac{x^2}{x+2}dx=\int\dfrac{\left(t-2\right)^2}{t}dt=\int\left(t-4+\dfrac{4}{t}\right)dt\)=\(\dfrac{t^2}{2}-4t+4ln\left|t\right|+C=\dfrac{x^2}{2}-2x+4ln\left|x+2\right|+C\).
Đặt t=x+2 \(\Rightarrow\) dt=dx.
\(\int\dfrac{x^2}{x+2}dx=\int\dfrac{\left(t-2\right)^2}{t}dt=\int\left(t-4+\dfrac{4}{t}\right)dt\)=\(\dfrac{t^2}{2}-4t+4ln\left|t\right|+C=\dfrac{x^2}{2}-2x+4ln\left|x+2\right|+C\).
1.\(\int\dfrac{\sin2x}{\cos^4x-4}dx\)
2.\(\int\sqrt{1-x^2}dx\)
3.\(\int\dfrac{xdx}{\sqrt{1+x^4}}dx\)
giúp mình với mn
1, \(\int\dfrac{x^3dx}{\left(x^8-4\right)^2}\)
2, \(\int\dfrac{2x+1}{x^4+2x^2+3x^2+2x-3}dx\)
3, \(\int\dfrac{sin\sqrt[3]{x}}{\sqrt[3]{x^2}}dx\)
4, \(\int\dfrac{dx}{sin^2x+2cos^2x}\)
5, \(\int\dfrac{sinx+cosx}{3+sin2x}dx\)
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) Tinh nguyen ham
a) A = \(\int\dfrac{x}{\sqrt{x+2}}.dx\) b) B = \(\int\dfrac{sinx+cosx}{\sqrt[3]{1-sin2x}}.dx\)
\(\int\dfrac{1}{cosx.cos\left(x+\dfrac{\pi}{4}\right)}dx\)
\(\int\dfrac{1}{x^3\left(1+x^2\right)}dx=\dfrac{a}{x^2}+blnx+cln\left(1+x^2\right).S=a+b+c=?\)
\(\int\dfrac{5-3x}{\left(x^2-5x+6\right)\left(x^2-2x+1\right)}dx=\dfrac{a}{x-1}-ln\left(\dfrac{x-b}{x-c}\right)+C.P=2a+b\)
Tìm nguyên hàm của các hàm số sau:
a) \(\int\left(6x-\dfrac{1}{sin^2x}+1\right)dx\)
b) \(\int\dfrac{x^3+2x^2-1}{x^2}dx\)
Tính :
\(I_1=\int\dfrac{x^2-2x+2}{\sqrt{x^2-2x}}dx\)
\(I_2=\dfrac{dx}{\left(x-1\right)\sqrt{x^2-2x+2}}\)
\(I_3=\dfrac{dx}{\left(2x+1\right)\sqrt{x^2-2x+2}}\)
Tính các nguyên hàm sau bằng phương pháp đổi biến số :
a) \(\int x^2\sqrt[3]{1+x^3}dx\) với \(x>-1\) (đặt \(t=1+x^3\))
b) \(\int xe^{-x^2}dx\) (đặt \(t=x^2\))
c) \(\int\dfrac{x}{\left(1+x^2\right)^2}dx\) (đặt \(t=1+x^2\))
d) \(\int\dfrac{1}{\left(1-x\right)\sqrt{x}}dx\) (đặt \(t=\sqrt{x}\))
e) \(\int\sin\dfrac{1}{x}.\dfrac{1}{x^2}dx\) (đặt \(t=\dfrac{1}{x}\))
g) \(\int\dfrac{\left(\ln x\right)^2}{x}dx\) (đặt \(t=\ln x\))
h) \(\int\dfrac{\sin x}{\sqrt[3]{\cos^2x}}dx\) (đặt \(t=\cos x\) )
i) \(\int\cos x\sin^3xdx\) (đặt \(t=\sin x\))
k) \(\int\dfrac{1}{e^x-e^{-x}}dx\) (đặt \(t=e^x\) )
l) \(\int\dfrac{\cos x+\sin x}{\sqrt{\sin x-\cos x}}dx\) (đặt \(t=\sin x-\cos x\))
mọi người tính giúp em mấy cái nguyên hàm với :
1, \(\int\dfrac{x^2-1}{x^4+1}dx\)
2, \(\int\dfrac{dx}{3x^2-3}\)
3, \(\int\dfrac{dx}{1+sinx}\)
4, \(\int\dfrac{sinxdx}{\sqrt{cos2x}}\)
5, \(\int\dfrac{xdx}{\sqrt[3]{1-3x}}\)
6, \(\int2^{2x}.3^{3x}.4^xdx\)
7,\(\int\dfrac{dx}{xlnx.ln\left(lnx\right)}\)
8, \(\int\dfrac{dx}{sinx.cos^3x}\)
9, \(\int\dfrac{cos2xdx}{sin^2x.cos^2x}\)
10, \(\int\dfrac{dx}{\sqrt{2x}-\sqrt{2x+1}}\)