Indefinite integration and its applications：練習問題

Evaluate the indefinite integral:
$$\int \left( \frac{3x^4 - \sqrt{x}}{x^2} + \frac{2}{x} \right) \, dx$$

Evaluate the indefinite integral: \(\int \left(x - \frac{1}{x}\right)^2 dx\)

Find the indefinite integral of the function \(f(x) = x^3(x^2+1)^5\) with respect to \(x\).

Evaluate the indefinite integral: \(\int \frac{3x^4 - 2x^2 + 5}{x^2} dx\)

The gradient of a curve \(y=f(x)\) at any point \((x,y)\) is given by \(\frac{dy}{dx} = 3x^2 - 4x + 1\). If the tangent to the curve at \(x=2\) is parallel to the line \(y = x\), and the curve passes through \((1, 5)\), find \(f(x)\).

Find the indefinite integral \(\int (4x^3 - \frac{2}{x} + 5) \, dx\).

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

The rate of change of the population of a town is modeled by \(\frac{dP}{dt} = 100e^{0.2t} + 50\), where \(t\) is the time in years. If the initial population is 5000, find the population function \(P(t)\).

The rate of change of a quantity $$Q$$ with respect to time $$t$$ is given by $$\frac{dQ}{dt} = 5t^4 - \frac{3}{t} + 2e^t$$.

(a) Find the general expression for $$Q(t)$$.

(b) If $$Q(1) = 2e$$, find the particular expression for $$Q(t)$$.

A particle moves along a straight line such that its acceleration a m/s$$^2$$ at time t seconds is given by
$$a(t) = 6t - \frac{1}{(t+1)^2} + 3e^{2t}$$
for $$t \ge 0$$.
(a) Given that the initial velocity of the particle is $$5$$ m/s, find the velocity function $$v(t)$$ in terms of $$t$$.
(b) If the initial displacement of the particle is $$2$$ m from the origin, find the displacement function $$s(t)$$ in terms of $$t$$.