Integration：練習問題

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

Find the exact area enclosed by the curve \(y = 4x - x^2\) and the x-axis.

The region \(R\) is bounded by the curve \(y = \sqrt{x}\), the x-axis, and the line \(x=4\). Find the volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the x-axis.

Find \(\int (6x^2 - 3x + 1) \, dx\).

The gradient of a curve is given by \(\frac{dy}{dx} = 3x^2 - 8x + 5\). If the curve passes through the point \((1, 4)\), find the equation of the curve.

Find \(\int (4x^3 - 6x^2 + 2) dx\).

Evaluate the definite integral \(\int_0^3 (3x^2 - 2x) \, dx\).

Find the exact area of the region bounded by the curve \(y = x^2 - 4x\) and the x-axis.

Find the indefinite integral of \(f(x) = 6x^2 - \frac{4}{x^3} + \sqrt{x}\) with respect to \(x\).

The gradient of a curve is given by \(\frac{dy}{dx} = 3x^2 - 8x + 5\). The curve passes through the point \((2, 1)\). Find the equation of the curve.