Integration：練習問題

Evaluate the definite integral \( \int_{0}^{\pi/4} \sin^2 x \, dx \).

Solve the differential equation \( \frac{dy}{dx} = y \cot x \) given that \( y = 5 \) when \( x = \frac{\pi}{2} \).

Find the finite area of the region bounded by the curves \( y = x^2 - 4x \) and \( y = 2x - x^2 \).

Use the substitution \( u = x - 1 \) to find the indefinite integral \( \int \frac{x}{\sqrt{x-1}} \, dx \).

Use integration by parts to find the indefinite integral \( \int x e^{2x} \, dx \).

Figure 2 shows the finite region \(R\) bounded by the curve \(y = \sqrt{\sin x}\), the \(x\)-axis, and the lines \(x = 0\) and \(x = \frac{\pi}{2}\). Find the exact volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

Use the substitution \(u = \sqrt{x}\) to evaluate the exact value of the definite integral \(\int_1^9 \frac{1}{x+\sqrt{x}} dx\).