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

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 các tích phân sau đây :

a) \(\int\limits^{\dfrac{\pi}{2}}_0\left(x+1\right)\cos\left(x+\dfrac{\pi}{2}\right)dx\)

b) \(\int\limits^1_0\dfrac{x^2+x+1}{x+1}\log_2\left(x+1\right)dx\)

c) \(\int\limits^1_{\dfrac{1}{2}}\dfrac{x^2-1}{x^4+1}dx\) (đặt \(t=x+\dfrac{1}{x}\) )

d) \(\int\limits^{\dfrac{\pi}{2}}_0\dfrac{\sin3xdx}{3+4\sin x-\cos2x}dx\)

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

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

Tính :

a) \(\int\limits^{\dfrac{\pi}{2}}_0\cos2x.\sin^2dx\)

b) \(\int\limits^1_{-1}\left|2^x-2^{-x}\right|dx\)

c) \(\int\limits^2_1\dfrac{\left(x+1\right)\left(x+2\right)\left(x+3\right)}{x^2}dx\)

d) \(\int\limits^2_0\dfrac{1}{x^2-2x-3}dx\)

e) \(\int\limits^{\dfrac{\pi}{2}}_0\left(\sin x+\cos x\right)^2dx\)

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

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) \(\int\dfrac{\sin x\cos x}{\sqrt{a^2\sin^2x+b^2\cos^2x}}dx\) (\(a^2\ne b^2\))