∫ calculus

1. **Problem Statement:** Differentiate the integral $$\int_0^x \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a}$$ under the integral sign to find the value of $$\int_

1. **State the problem:** We are given a circle with radius $r$ increasing at a rate of $\frac{dr}{dt} = 5$ cm/s. We need to find the rate at which the area $A$ of the circle is in

1. **State the problem:** We are given the function $$y = (x^2 + 1)\sqrt{2x - 3}$$.

1. **State the problem:** We want to find the indefinite integral $$\int \sqrt{\tan x} \, dx$$. 2. **Recall the formula and substitution:** Integrals involving square roots of trig

1. **Stating the problem:** We want to evaluate the integral $$\int_0^\infty e^{-ax} \sin(\beta x) \, dx$$ where $a > 0$ and $\beta$ are constants. 2. **Formula and important rules

1. The problem asks to create a table of 5 numbers where the function values are increasing and decreasing as $x$ approaches 1. 2. We will choose values of $x$ approaching 1 from b

1. The problem is to understand and create a table showing where a function is increasing or decreasing. 2. A function is increasing on intervals where its derivative is positive (

1. **State the problem:** Find the limit as $x$ approaches 7 of the function $$\frac{5x^2 - 7x + 2}{x^2 - 1}.$$\n\n2. **Recall the limit rule for rational functions:** If the funct

1. The problem is to understand what decreasing and increasing values mean in mathematics. 2. A function is called increasing on an interval if for any two numbers $x_1$ and $x_2$

1. **State the problem:** Find the limit as $x$ approaches 7 of the function $$\frac{5x^2 - 7x + 2}{x^2 - 1}$$. 2. **Recall the limit rule:** If the function is continuous at $x=7$

1. **State the problem:** Find $\frac{dy}{dx}$ by implicit differentiation for the equation $$2x^2 + xy - y^2 = 2.$$\n\n2. **Recall the rules:** When differentiating implicitly, tr

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$2x^3 + xy - y^2 = 2.$$\n\n2. **Recall the formula and rules:** When differe

1. **Problem Statement:** We are given the function $$f(x) = x^4 + \frac{1}{2}x^3 - 5x^2 + \tan\left(\frac{x}{2}\right)$$ and asked to find at which of the given points $$x = -2, -

1. **Problem Statement:** We are given a function $f$ defined on the interval $[a,b]$ with $f(b) > f(a)$ and $a \leq x \leq b$. The derivative $f'(x)$ exists for all $x$ in $(a,b)$

1. **Problem Statement:** We are given a function $f$ defined on the interval $[a,b]$ with $f(b) > f(a)$ and the derivative $f'(x)$ exists for all $x$ in $(a,b)$ except at $x=0$. W

1. **Problem statement:** Calculate $\frac{dz}{dt}$ for the function $$z = f(x,y) = \sqrt{x^2 - y^2}$$ where $$x = e^{2t}$$ and $$y = e^{-t}$$. 2. **Formula and rules:** To find $\

1. **State the problem:** Find the volume of the solid obtained by rotating the region bounded by the curves $y = x^2 + 8x + 7$ and $y = 0$ about the x-axis. 2. **Identify the regi

1. **Problem Statement:** Expand $\log_e x$ in powers of $(x-1)$ and hence evaluate $\log_e 1.1$ correct up to 4 decimals. 2. **Formula and Explanation:** The Taylor series expansi

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\cot x}{1 + \csc x}$$. 2. **Recall formulas and rules:**

1. The problem is to find the derivative of a function, but the function is not specified in the question. 2. To find the derivative of a function $f(x)$, we use the definition of

1. **Problem Statement:** Find the Taylor series expansions of the following functions at the given points $a$. 2. **Recall:** The Taylor series of a function $f(x)$ at $x=a$ is gi