∫ calculus

1. The problem asks to find the value or expression when the integral (Int) equals 4. 2. To proceed, we need the specific integral expression or function to evaluate.

1. **State the problem:** We want to prove using the definition of convergence that $$\lim_{n \to \infty} \frac{5}{1+n^2} = 0.$$\n\n2. **Recall the definition of convergence:** A s

1. The problem is to evaluate the limit: $$\lim_{n \to \infty} \frac{5}{1+n^2}$$. 2. The formula for limits involving rational functions as $n$ approaches infinity is to analyze th

1. **Problem Statement:** Determine the limits of the function $f(x)$ as $x$ approaches $-1$, $1^-$, and $1^+$ using the graph.

1. **Problem statement:** Prove that $$\lim_{n \to \infty} \frac{2n}{2+n} = 2$$ using the definition of convergence. 2. **Definition of convergence:** A sequence $$a_n$$ converges

1. সমস্যাটি হলো: আলাদা আলাদা না করে একসাথে যোগ করতে হবে লেইবনিটজের নিয়ম ব্যবহার করে। 2. লেইবনিটজের নিয়ম বলে যে, যদি দুটি ফাংশনের গুণফল থাকে $u(x)\cdot v(x)$, তবে তার ডেরিভেটিভ হব

1. The problem is to find the second derivative of the function $f(x) = x^3$. 2. Recall that the first derivative of a function $f(x)$, denoted $f'(x)$ or $\frac{d}{dx}f(x)$, gives

1. **State the problem:** Find the derivative of the function $f(x) = x^2$. 2. **Recall the power rule for derivatives:** If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. **State the problem:** Differentiate the function $$f(x) = x^3 - 4x + 6$$ with respect to $$x$$. 2. **Recall the differentiation rules:**

1. **State the problem:** We need to find the limit as $x$ approaches 5 of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{0 + 2x}.$$ 2. **Rewrite the expression:** The expressio

1. **State the problem:** Find the limit $$\lim_{x \to \frac{\pi}{2}} \frac{3 \cos x + \cos 3x}{\left(\frac{\pi}{2} - x\right)^3}$$

1. **Problem statement:** Find the derivative of the function $$f(x) = \tan^4 \left( \sin^2 \left( x^3 + 2x \right) \right).$$ 2. **Formula and rules:** We will use the chain rule

1. **Problem (a):** Show that if the series $\sum_{n=1}^\infty a_n$ converges, then $\lim_{n \to \infty} a_n = 0$ using partial sums. 2. **Formula and explanation:** Let $S_n = \su

1. **State the problem:** Find the second derivative of the function $$f(x)=\frac{4e^{2x}+3x}{e^{x^{2}}}.$$\n\n2. **Rewrite the function:** Simplify the expression by writing it as

1. **State the problem:** Differentiate the function $$f(x) = \frac{e^{x^3} - 1}{x}$$ with respect to $$x$$. 2. **Formula used:** We will use the quotient rule for differentiation,

1. **State the problem:** We are given a piecewise function $$f(x) = \begin{cases} x^3 + 2, & x > 0 \\ 3 - 2x^2, & x < 0 \end{cases}$$

1. **State the problem:** Find the equation of the tangent line to the curve of the function $f(x) = e^{2x+1}$ at the point $\left(-\frac{1}{2}, 1\right)$. 2. **Recall the formula

1. We are given the function $f(x) = e^x \cdot g(x)$, with $g(0) = 2$ and $g'(0) = 1$. We need to find $f'(0)$. 2. To find the derivative of a product of two functions, we use the

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

1. **Stating the problem:** Find the area inside both polar curves $r=3+2\cos(\theta)$ and $r=3+2\sin(\theta)$.\n\n2. **Formula for area inside a polar curve:** The area enclosed b

1. **State the problem:** We need to find the limit $$\lim_{x \to -3} (f(x) + h(x))$$ where functions $f$ and $h$ are given graphically. 2. **Recall the limit sum rule:** The limit