Integration：練習問題

Evaluate the indefinite integral \(\int \left(3x^2 + \frac{2}{x^3} + 5\sqrt{x}\right) dx\).

Using the substitution \(u = \ln x\) , evaluate the definite integral \(I = \int_{e}^{e^2} \frac{1}{x \ln x} dx\) .

Evaluate the definite integral \(\int_1^4 \left(3\sqrt{x} + \frac{1}{x^2}\right) dx\).

Find the exact value of the definite integral:

$$\int_3^4 \frac{2x^2 + 3x + 1}{x(x+1)} dx$$

Evaluate the definite integral \(\int_1^4 \left(\frac{x^3 + 2}{\sqrt{x}}\right) dx\).

Part (a)

Find the indefinite integral:

\[\int \left( 4x^3 - \frac{6}{x^2} + \sqrt{x} \right) dx\]

Part (b)

Given that \(\frac{dy}{dx} = 3x^2 - 4x + 1\), and that the curve passes through the point \((2, 5)\), find the equation of the curve \(y\) in terms of \(x\).

The curve $C$ has the equation $y = \frac{4x^2 + 3x - 1}{\sqrt{x}}$, for $x > 0$. The line $L$ has the equation $y = 7x - 1$.

Part (a)

Show that the equation for the curve $C$ can be written in the form $y = 4x^{3/2} + 3x^{1/2} - x^{-1/2}$.

Part (b)

Find the exact coordinates of the point of intersection between the curve $C$ and the line $L$.

Part (c)

Find the exact area of the finite region bounded by the curve $C$ and the line $L$.