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, \(\int\)\(\dfrac{dx}{\sqrt{1-x^2}}\)

Tính :

a) \(\int\limits^2_{-1}\left(5x^2-x+e^{0,5x}\right)dx\)

b) \(\int\limits^2_{0,5}\left(2\sqrt{x}+\dfrac{3}{x^2}+\cos x\right)dx\)

c) \(\int\limits^2_1\dfrac{dx}{\sqrt{2x+3}}\) (đặt \(t=\sqrt{2x+3}\) ) 

d) \(\int\limits^2_1\sqrt[3]{3x^3+4}x^2dx\)  (đặt \(t=\sqrt[3]{3x^3+4}\) )

e) \(\int\limits^2_{-2}\left(x-2\right)\left|x\right|dx\)

g) \(\int\limits^0_1x\cos xdx\)

h) \(\int\limits^{\dfrac{\pi}{2}}_{\dfrac{\pi}{6}}\dfrac{1+\sin2x+\cos2x}{\sin x+\cos x}dx\)

i) \(\int\limits^{\dfrac{\pi}{2}}_0e^x\sin xdx\)

k) \(\int\limits^e_1x^2\ln^2xdx\)

Tính các tích phân sau :

a) \(\int\limits^4_{-2}\left(\dfrac{x-2}{x+3}\right)^2dx\) (đặt \(t=x+3\) )

b) \(\int\limits^6_{-4}\left|x+3\right|-\left|x-4\right|dx\)

c) \(\int\limits^2_{-3}\dfrac{dx}{\sqrt{x+7}+3}\) (đặt \(t=\sqrt{x+7}\) hoặc \(t=\sqrt{x+7}+3\) )

d) \(\int\limits^{\dfrac{\pi}{2}}_0\dfrac{\cos x}{1+4\sin x}dx\)

e) \(\int\limits^2_1\dfrac{x^9}{x^{10}+4x^5+4}dx\) (đặt \(t=x^5\) )

g) \(\int\limits^3_0\left(x+2\right)e^{2x}dx\) 

h) \(\int\limits^5_2\dfrac{\sqrt{4+x}}{x}dx\) (đặt \(t=\sqrt{4+x}\) )

Tính các tích phân sau

1.I=\(\int\limits^{\frac{\Pi}{4}}_0\) (x+1)sin2xdx

2.I=\(\int\limits^2_1\frac{x^2+3x+1}{x^2+x}dx\)

3.I=\(\int\limits^2_1\frac{x^2-1}{x^2}lnxdx\)

4. I=\(\int\limits^1_0x\sqrt{2-x^2}dx\)

5.I=\(\int\limits^1_0\frac{\left(x+1\right)^2}{x^2+1}dx\)

6. I=\(\int\limits^5_1\frac{dx}{1+\sqrt{2x-1}}\)

7. I=\(\int\limits^3_1\frac{1+ln\left(x+1\right)}{x^2}dx\)

8.I=\(\int\limits^1_0\frac{x^3}{x^4+3x^2+2}dx\)

9. I=\(\int\limits^{\frac{\Pi}{4}}_0x\left(1+sin2x\right)dx\)

10. I=\(\int\limits^3_0\frac{x}{\sqrt{x+1}}dx\)

Hãy chỉ ra kết quả nào dưới đây đúng :

a) \(\int\limits^{\dfrac{\pi}{2}}_0\sin xdx+\int\limits^{\dfrac{3\pi}{2}}_{\dfrac{\pi}{2}}\sin xdx+\int\limits^{2\pi}_{\dfrac{3\pi}{2}}\sin xdx=0\)

b) \(\int\limits^{\dfrac{\pi}{2}}_0\left(\sqrt[3]{\sin x}-\sqrt[3]{\cos x}\right)dx=0\)

c) \(\int\limits^{\dfrac{1}{2}}_{-\dfrac{1}{2}}\ln\dfrac{1-x}{1+x}dx=0\)

d) \(\int\limits^2_0\left(\dfrac{1}{1+x+x^2+x^3}+1\right)dx=0\)

Tính các tích phân sau bằng phương pháp đổi biến số :

a) \(\int\limits^2_1x\left(1-x\right)^5dx\) (đặt \(t=1-x\))

b) \(\int\limits^{\ln2}_0\sqrt{e^x-1}dx\) (đặt \(t=\sqrt{e^x-1}\))

c) \(\int\limits^9_1x\sqrt[3]{1-x}dx\) (đặt \(t=\sqrt[3]{1-x}\) )

d) \(\int\limits^1_{-1}\dfrac{2x+1}{\sqrt{x^2+x+1}}dx\) (đặt \(u=\sqrt{x^2+x+1}\))

e) \(\int\limits^2_1\dfrac{\sqrt{1+x^2}}{x^4}dx\) (đặt \(t=\dfrac{1}{x}\) )