∫ calculus

1. **โจทย์:** กำหนดให้ $f(x) = x^2 + 3x - 4$, $g(x) = 5 - 6x$ และ $h(x) = 7x + 8$ จงหา

1. **State the problem:** We want to find the derivative of the function $$f(x) = 3^x$$ with respect to $$x$$. 2. **Recall the formula:** The derivative of an exponential function

1. **State the problem:** We want to differentiate the function $$f(x) = 4e^{7x}$$ with respect to $$x$$. 2. **Recall the differentiation rule for exponential functions:** The deri

1. **State the problem:** We need to evaluate the definite integral $$\int_{1}^{10} \frac{2624}{x^{2} - x - 1} \, dx.$$ 2. **Rewrite the integral:** The integrand is $$\frac{2624}{

1. **State the problem:** We are given the curve $$y = x^3 - 6x^2 + 9x + 4$$ and need to find its derivative $$\frac{dy}{dx}$$. 2. **Recall the derivative rules:**

1. **Problem Statement:** We are asked to analyze the functions $y = e^{-x}$ and $y = \sin x$ on the interval $0 \leq x \leq \pi$.

1. **Problem statement:** Calculate the following limits and find the asymptotes of given curves. ---

1. **State the problem:** Find the limit

1. **State the problem:** We need to find the limit \(\lim_{x \to -5} \frac{x^3 + 5x^2 + x + 5}{x^2 + 12x + 35}\). 2. **Identify the form:** Substitute \(x = -5\) directly:

1. **State the problem:** We want to find the limit $$\lim_{x \to \pm \infty} \left( \sqrt{3x^2+1} - \sqrt{2x^2+x} \right).$$ 2. **Recall the formula and technique:** When dealing

1. **State the problem:** We want to find the integral $$\int x \ln(3x) \, dx.$$\n\n2. **Recall the integration by parts formula:** $$\int u \, dv = uv - \int v \, du.$$\n\n3. **Ch

1. **State the problem:** We are given the rate of change of the value of a house as $\frac{dV}{dt}=150e^{0.1t}$ and the initial value $V(0)=60$ (since 60,000 Ghana cedis is 60 tho

1. **State the problem:** We need to find the indefinite integral $$\int (x^{23} - 3x^{x} + 3x^{2x} - 5x^{2x}) \, dx$$. 2. **Analyze each term:**

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{1 + \sin x - \cos x}{1 - \sin x - \cos x}$$.

1. **State the problem:** We are given the function $$z = \cos(x) \cdot \ln(y) + 2y$$ where $$x = |s| + e^t$$ and $$y = s^2 + 3t$$.

1. **Problem Statement:** Evaluate the triple integrals in part (a), find the volume in part (b), and convert the integral in part (c) to cylindrical coordinates.

1. **State the problem:** We want to find the indefinite integral $$\int \frac{1-\sqrt{1+e^x}}{\sqrt[3]{1+e^x}}\,dx.$$ 2. **Identify substitution:** Let $$u = 1 + e^x.$$ Then, $$du

1. **Problem Statement:** Evaluate the triple integrals in part (a), find the volume in part (b), and convert the integral in part (c) to cylindrical coordinates.

1. **Problem Statement:** (a) Find the inverse Laplace transform of $$C(f) = \frac{2(e^{-s} - e^{-3s})}{s^2 - 4}$$.

1. **Problem Statement:** Find the limit of a function involving trigonometric expressions as the variable approaches a certain value. 2. **General Formula and Rules:**

1. সমস্যাটি হলো $y = x^2$ এবং $x = y^2$ দ্বারা সীমাবদ্ধ ক্ষেত্রের ক্ষেত্রফল নির্ণয় করা। 2. প্রথমে সমীকরণগুলো লিখি: