Bài 2: Tích phân

Tích phân \(\int\limits^{\pi}_0\cos^3x.\sin x\text{d}x\) bằng

1. \(-\dfrac{1}{4}\pi^4\).
2. \(-\pi^4\).
3. \(0\).
4. \(-\dfrac{1}{4}\).

Tích phân \(\int\limits^{\frac{\pi}{2}}_0\sin\left(\dfrac{\pi}{4}-x\right)\text{d}x\) bằng

1. \(\sqrt{2}\).
2. \(0\).
3. \(-\sqrt{2}\).
4. \(1\).

Tích phân  \(\int\limits^2_{\frac{1}{2}}\dfrac{1-3x}{\left(x+1\right)^2}\text{d}x\) bằng

1. \(\dfrac{4}{3}-3\ln2\).
2. \(3\ln2\).
3. \(\dfrac{4}{3}\).
4. \(-\dfrac{4}{3}+3\ln2\).

Tích phân \(\int\limits^2_{\frac{1}{2}}\dfrac{1}{x\left(x+1\right)}\text{d}x\) bằng

1. \(1\).
2. \(2\).
3. \(\ln2\).
4. \(\ln3\).

Tích phân  \(\int\limits^2_0\left|1-x\right|\text{d}x\) bằng

1. \(\dfrac{1}{2}\).
2. \(1\).
3. \(\dfrac{3}{2}\).
4. \(2\).

Tích phân    \(\int\limits^1_0\sqrt{1-x^2}\text{d}x\) bằng

1. \(\dfrac{\pi}{3}\).
2. \(\dfrac{\pi}{4}\).
3. \(\dfrac{\pi}{6}\).
4. \(1\).

Tích phân \(\int\limits^{\dfrac{\pi}{2}}_0\left(x+1\right)\sin x\text{d}x\) bằng

1. \(1\).
2. \(2\).
3. \(\pi\).
4. \(\dfrac{\pi}{2}\).

Tích phân  \(\int\limits^1_0\ln\left(1+x\right)\text{d}x\)  bằng

1. \(2\ln2-1\).
2. \(2\ln2+1\).
3. \(1\).
4. \(\ln2\).

Tích phân  \(\int\limits^e_1x^2\ln x\text{d}x\) bằng

1. \(\dfrac{1}{3}\).
2. \(\dfrac{2}{9}e^3\).
3. \(\dfrac{1}{3}e^3\).
4. \(\dfrac{2}{9}e^3+\dfrac{1}{9}\).

Tích phân  \(\int\limits^1_0\left(1+3x\right)^{\dfrac{3}{2}}\text{d}x\) bằng

1. \(\dfrac{62}{15}\).
2. \(4\).
3. \(\dfrac{62}{5}\).
4. \(\dfrac{155}{2}\).

Cho \(a< b< c\), \(\int\limits^b_af\left(x\right)\text{d}x=5\), \(\int\limits^b_cf\left(x\right)\text{d}x=2\) . Tích phân \(\int\limits^c_af\left(x\right)\text{d}x\) bằng

1. \(-2\).
2. \(3\).
3. \(8\).
4. \(0\).

Giải phương trình ẩn \(a\) sau đây : \(\int\limits^a_0\cos x\text{d}x=0\).

1. \(a=\dfrac{\pi}{3}\).
2. \(a=\frac{\pi}{3}+k2\pi,k\in\mathbb{Z}\).
3. \(a=\frac{\pi}{6}+k2\pi,k\in\mathbb{Z}\).
4. \(a=k\pi,k\in\mathbb{Z}\).

Giá trị của \(\sin\left(\int\limits_0^{\pi}\pi x\cos x\text{d}x\right)\)  là

1. \(1\).
2. \(0\).
3. \(\pi\).
4. \(\dfrac{\sqrt{3}}{2}\).

Tích phân \(\int\limits^1_0\left(x-2\right)e^{2x}\text{d}x\) bằng

1. \(\dfrac{5+3e^2}{4}\).
2. \(\dfrac{-5-3e^2}{4}\).
3. \(\dfrac{5-3e^2}{4}\).
4. \(\dfrac{5-3e^2}{2}\).

Tích phân \(\int\limits^{\frac{\pi}{2}}_0\sin^5x\text{d}x\)  bằng

1. \(\dfrac{1}{5}\).
2. \(\dfrac{2}{15}\).
3. \(\dfrac{8}{15}\).
4. \(\dfrac{4}{15}\).

Tích phân  \(\int\limits^{\ln5}_0\dfrac{e^x\sqrt{e^x-1}}{e^x+3}\text{d}x\) bằng

1. \(4-\pi\).
2. \(4+\pi\).
3. \(2-\pi\).
4. \(2+\pi\).

Tích phân  \(\int\limits^{\dfrac{\pi}{3}}_{\dfrac{\pi}{6}}\dfrac{\ln\left(\sin x\right)}{\cos^2x}\text{d}x\) bằng

1. \(\dfrac{2}{\sqrt{3}}\ln\dfrac{3}{4}-\dfrac{\pi}{6}\).
2. \(\dfrac{1}{\sqrt{3}}\ln\dfrac{3\sqrt{3}}{4}-\dfrac{\pi}{6}\).
3. \(\dfrac{\sqrt{3}}{2}\ln\dfrac{3}{4}+\dfrac{\pi}{6}\).
4. \(\dfrac{2}{\sqrt{3}}\ln\dfrac{3\sqrt{3}}{4}+\dfrac{\pi}{6}\).

Tích phân  \(\int\limits^1_0\dfrac{\text{d}x}{\sqrt{4-x^2}}\) bằng

1. \(\dfrac{\pi}{3}\).
2. \(\dfrac{\pi}{4}\).
3. \(\dfrac{\pi}{6}\).
4. \(\dfrac{\pi}{8}\).

Biết \(\int\limits^4_3\dfrac{\text{d}x}{x^2+x}=a\ln2+b\ln3+c\ln5\), với \(a,b,c\) là các số nguyên. Tổng \(a+b+c\) bằng

1. \(6\).
2. \(2\).
3. \(-2\).
4. \(0\).

Tích phân  \(\int^{\dfrac{\pi}{2}}_0\left(2x-1-\sin x\right)\text{d}x\) bằng

1. \(\dfrac{\pi^2}{4}-\dfrac{\pi}{2}-1\).
2. \(\dfrac{\pi^2}{4}-\dfrac{\pi}{2}+1\).
3. \(\dfrac{\pi^2}{4}+\dfrac{\pi}{2}-1\).
4. \(-\dfrac{\pi^2}{4}+\dfrac{\pi}{2}-1\).