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 :

a) \(\int\limits^1_0\left(y^3+3y^2-2\right)dy\)

b) \(\int\limits^4_1\left(t+\dfrac{1}{\sqrt{t}}-\dfrac{1}{t^2}\right)dt\)

c) \(\int\limits^{\dfrac{\pi}{2}}_0\left(2\cos x-\sin2x\right)dx\)

d) \(\int\limits^1_0\left(3^s-2^s\right)^2ds\)

e) \(\int\limits^{\dfrac{\pi}{3}}_0\cos3xdx+\int\limits^{\dfrac{3\pi}{2}}_0\cos3xdx+\int\limits^{\dfrac{5\pi}{2}}_{\dfrac{3\pi}{2}}\cos3xdx\)

g) \(\int\limits^3_0\left|x^2-x-2\right|dx\)

h) \(\int\limits^{\dfrac{5\pi}{4}}_{\pi}\dfrac{\sin x-\cos x}{\sqrt{1+\sin2x}}dx\)

i) \(\int\limits^4_0\dfrac{4x-1}{\sqrt{2x+1}+2}dx\)