Integration：练习题

Calculate the area under the curve \(y = \frac{1}{x}\) between \(x = 1\) and \(x = e\).

Using integration by parts, evaluate \(\int x \cos(x) \, dx\).

Find the indefinite integral \(\int (6x^2 - 4x + 3) \, dx\).

Evaluate the definite integral \(\int_{0}^{2} (3x + 1)^2 \, dx\).

Find the integral \(\int \sin(2x) \, dx\).

Determine the exact value of the definite integral \(\int_{0}^{\frac{\pi}{2}} (2 \sin x + \cos x) dx\).

Find the definite integral of the function f(x) between the limits 1 and 3.
\(\int_{1}^{3} (4x^3 - 2x + 5) dx\)