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

Xét tích phân I=\(\int\limits^{\dfrac{\pi}{2}}_0\dfrac{sin2x}{\sqrt{1+cosx}}dx\). Nếu đặt t=\(\sqrt{1+cosx}\), khẳng định nào dưới đây là đúng?

A. I= \(\int\limits^1_{\sqrt{2}}\dfrac{4t^3-4t}{t}dt\)

B. I= \(\int\limits^1_{\sqrt{2}}\dfrac{-4t^3+4t}{t}dt\)

C. I= \(4\int\limits^{\sqrt{2}}_1\left(t^2-1\right)dt\)

D. I= \(-4\int\limits^{\sqrt{2}}_1\left(t^2-1\right)dt\)

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

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) \(\int\limits^{\dfrac{1}{2}}_{-\dfrac{1}{2}}\sqrt[3]{\left(1-x\right)^2dx}\)

b) \(\int\limits^{\dfrac{\pi}{2}}_0\sin\left(\dfrac{\pi}{4}-x\right)dx\)

c) \(\int\limits^2_{\dfrac{1}{2}}\dfrac{1}{x\left(x+1\right)}dx\)

d) \(\int\limits^2_0x\left(x+1\right)^2dx\)

e) \(\int\limits^2_{\dfrac{1}{2}}\dfrac{1-3x}{\left(x+1\right)^2}dx\)

g) \(\int\limits^{\dfrac{\pi}{2}}_{-\dfrac{\pi}{2}}\sin3x\cos5xdx\)

Áp dụng phương pháp tính tích phân, hãy tính các tích phân sau :

a) \(\int\limits^{\dfrac{\pi}{2}}_0x\cos2xdx\)

b) \(\int\limits^{\ln2}_0xe^{-2x}dx\)

c) \(\int\limits^1_0\ln\left(2x+1\right)dx\)

d) \(\int\limits^3_2\left|\ln\left(x-1\right)-\ln\left(x+1\right)\right|dx\)

e) \(\int\limits^2_{\dfrac{1}{2}}\left(1+x-\dfrac{1}{x}\right)e^{x+\dfrac{1}{x}}dx\)

g) \(\int\limits^{\dfrac{\pi}{2}}_0x\cos x\sin^2xdx\)

h) \(\int\limits^1_0\dfrac{xe^x}{\left(1+x\right)^2}dx\)

i) \(\int\limits^e_1\dfrac{1+x\ln x}{x}e^xdx\)