Question

[Thiên An]

Tính tích phân các hàm lượng giác sau :

a) \(I_1=\int_1^2\left(3x^2+\cos x+\frac{1}{x}\right)dx\)

b) \(I_2=\int_1^2\left(\frac{4}{x}-5x^2+2\sqrt{x}\right)dx\)

c) \(I_3=\int_a^b\frac{\left|x\right|}{x}dx\), với ab>0

d) \(I_5=\int_0^{\frac{\pi}{2a}}\left(x+3\right)\sin ax.dx\) với a>0

e)\(I_4=\int_0^{\pi}\sqrt{\frac{1+\cos2x}{2}}dx\)

Answer 1

[Đoàn Minh Trang]

\(I_1=3\int_1^2x^2dx+\int_1^2\cos xdx+\int_1^2\frac{dx}{x}=x^3\)\(|^2 _1\)+\(\sin x\)\(|^2_1\) +\(\ln\left|x\right|\)\(|^2_1\)

    \(=\left(8-1\right)+\left(\sin2-\sin1\right)+\left(\ln2-\ln1\right)\)

     \(=7+\sin2-\sin1+\ln2\)

Answer 2

[Đoàn Minh Trang]

b) \(I_2=4\int_1^2\frac{dx}{x}-5\int_1^2x^4dx+2\int_1^2\sqrt{x}dx\)

         \(=4\left(\ln2-\ln1\right)-\left(2^5-1^5\right)+\frac{4}{3}\left(2\sqrt{2}-1\sqrt{1}\right)\)

         \(=4\ln2+\frac{8\sqrt{2}}{3}-32\frac{1}{3}\)