Integration：練習問題

Evaluate the definite integral $$\int_1^2 (3x^2 - 4x + 1) dx$$.

Determine the general solution for the integral \(\int x^2 e^{2x} dx\).

Find the exact value of the definite integral \(\int_{0}^{\frac{\pi}{4}} \cos^2(x) dx\).

Using partial fractions, find the exact value of the integral \(\int_0^1 \frac{x}{(x+1)(x+2)} \, dx\).

Evaluate the definite integral \(\int_0^{\frac{\pi}{4}} \cos^2 x \, dx\).

The gradient of a curve is given by \(\frac{dy}{dx} = 6x^2 - 5\). If the curve passes through the point \((1, 4)\), find the equation of the curve.

Given that \(\frac{1}{x(x+1)} \equiv \frac{1}{x} - \frac{1}{x+1}\), find the exact value of \(\int_1^e \frac{1}{x(x+1)} \, dx\).

Calculate the exact volume generated when the region bounded by the curve \(y = \frac{2}{x}\), the x-axis, and the lines \(x=1\) and \(x=3\) is rotated completely about the x-axis.

The gradient of a curve at any point \((x, y)\) is given by \(\frac{dy}{dx} = 4x - 3x^{-\frac{1}{2}}\).

The curve passes through the point \(P(4, 8)\).

(a) Find the equation of the curve.

(b) Calculate the exact area of the region bounded by the curve, the x-axis, and the lines \(x=1\) and \(x=4\).

(a) Use the trapezium rule with 4 intervals to estimate the value of the integral \(I = \int_1^3 \frac{1}{\sqrt{x^3 + 1}} \, dx\). Give your answer correct to 3 significant figures.

(b) Find the exact value of \(\int_0^{\frac{\pi}{4}} \sin^2 x \, dx\).