∫ calculus

1. **State the problem:** We need to sketch a function $f$ continuous on $[1,5]$ with a second derivative on $(1,5)$ satisfying:

1. **State the problem:** We want to find the integral $$\int \frac{4x - 1}{(x + 1)(x + 2)} \, dx.$$\n\n2. **Use partial fraction decomposition:** Express the integrand as $$\frac{

1. The problem asks to find the points of inflection of $f$ and the intervals where $f$ is concave down, given the graph of its derivative $f'$. 2. Points of inflection of $f$ occu

1. The problem is to find the limit as $x \to \infty$ of the expression $$\frac{t^4 + 2x^5 - 3x^5}{x^5}$$. 2. First, divide each term in the numerator by the highest power of $x$ i

1. Problem (a): Differentiate $$y = \frac{x^3 + 3x}{(x+1)(x+2)}$$ 2. First, simplify the denominator:

1. **State the problem:** We have the curve defined by $$y = 4\sqrt{x} - \frac{x}{2} + 1$$ for $$0 \leq x \leq 64$$. We need to find values of $$a, b, c$$ at $$x=16, 32, 48$$ respe

1. **State the problem:** We are given the rate of change of cost per hour $P$ with respect to time $t$ as $$\frac{dP}{dt} = 20 - \frac{980}{t^2}, \quad 0 < t \leq 12.$$ We need to

1. **State the problem:** We have the function $f(x) = (6 - 3x)(4 + x)$ and a shaded region $R$ bounded by the x-axis, y-axis, and the graph of $f$. We need to find: (a) An integra

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = u^2 + 4$$ where $$\int \frac{du}{u^2 + 4} = \int dx$$. 2. **Rewrite the integral:** We have

1. **State the problem:** (a) Find the indefinite integral $\int (6x^2 - 3x) \, dx$.

1. **State the problem:** We need to find the indefinite integral $$\int \frac{x - 5}{(x - 9)(x + 3)} \, dx + c$$. 2. **Use partial fraction decomposition:** Express the integrand

1. **Problem Statement:** Check if each function $u(x,y)$ is homogeneous and find its degree if yes.

**Problem:** Calculate the following integrals step-by-step. 1. \( \int x^3 \, dx \)

1. **State the problem:** Find the equation of the normal line to the graph of the function $$F(x) = (x^4 - 3x^2 + 2x)(x^3 - 2x + 3)$$

1. **Problem statement:** Differentiate each of the given functions: a) $y = \sqrt[3]{\frac{x+1}{x-1}}$

1. The problem states the formula for $m^{(t)}$ as given: $$m^{(t)} = \frac{\int x f^{(x)} \, dx}{\int f^{(x)} \, dx} - t = \frac{\int F^{(x)} \, dx}{\int f^{(x)} \, dx} = \frac{F^

1. **State the problem:** Find the equations of the tangent and normal lines to the graph of the function $$F(x) = x^2 + 5x$$ at the point where $$x = -2$$. 2. **Find the point on

1. The problem is to understand and extract the formula for $m(t)$ given the integral expressions involving $\hat{f}(x)$ and $\hat{F}(x)$.\n\n2. The first expression is \n$$m(t) =

1. **State the problem:** Differentiate the function $$y = \frac{e^{-4x}}{4 e^{4x}}$$ with respect to $$x$$. 2. **Simplify the function:**

1. **State the problem:** We are given a function $f$ with vertical asymptotes at $x=0$ and $x=4$, and a horizontal asymptote at $y=-2$. We need to determine which of the given lim

1. **State the problem:** We need to find the limit $$\lim_{x \to \frac{\pi}{4}} h(x)$$ where $$h(x) = \frac{2\sin(x)}{1 - \cos(2x)}.$$\n\n2. **Step A: Direct substitution.** Subst