∫ calculus

1. **State the problem:** We are given the function $g(x) = x^2 e^{4x}$ and need to find intervals of concavity, points of inflection, local extrema, and determine if there are any

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{\sqrt{4x+8} - 6}{x^2 - 5x + 6}$$. 2. **Identify the indeterminate form:** Substitute $x=2$ directly:

1. Problem: Find the limit $$\lim_{x \to 1} \frac{x^2 - 3x + 2}{x^2 - 4x + 3}$$. 2. Factor numerator and denominator:

1. Problem: Naći jednačinu tangente na krivu $y + \cos^2 x y = \frac{9}{2}$ u tački $M(\pi, 4)$.\n\n2. Prvo, treba da diferenciramo implicitno jednačinu da bismo dobili $\frac{dy}{

1. **Problem Statement:** We are given the graph of a function and need to find the open intervals where the function is increasing, the open intervals where it is decreasing, and

1. **State the problem:** We need to evaluate the integral $$\int \frac{x + 3}{2x^3 - 8x^2} \, dx.$$\n\n2. **Simplify the denominator:** Factor the denominator:\n$$2x^3 - 8x^2 = 2x

1. Το πρόβλημα ζητά να βρούμε το gradient της συνάρτησης $$f(x,y)=3x^2+2xy+y^2$$. 2. Το gradient μιας συνάρτησης δύο μεταβλητών $$f(x,y)$$ είναι το διάνυσμα των μερικών παραγώγων τ

1. **Stating the problem:** We are given the function describing the depth of fluid in a tank as a function of time $t$ hours: $$y = 5 \left(2 - \frac{t}{15}\right)^2$$

1. نبدأ بكتابة المعطيات: القوة \( ق = ٣ س^٢ + ل ك ص \) ومعادلة الخط \( ٢ س - ص = ٤ \) والنقطة \( ب(٤, -١) \). 2. الهدف هو إيجاد \( جـ \) حيث \( جـ = \nabla ق \) (متجه التدرج) عند ا

1. نبدأ ببيان المسألة: نريد حساب ميل المماس للمنحنى المعطى بالعلاقة $$y=3x-1$$ عند النقطة التي يكون فيها $$y=\frac{\pi}{4}$$. 2. المعادلة المعطاة هي دالة خطية من الشكل $$y=3x-1$$.

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{x^3}{x}$$. 2. **Simplify the expression:** Since $x \neq 0$ in the limit process (we approach 0 but never

1. **Problem 1:** Find the limit $$\lim_{x \to 0} \left(1 + \frac{1}{x}\right)^x$$. 2. **Step 1:** Recognize that as $$x \to 0$$, the expression inside the parentheses $$1 + \frac{

1. **Problem:** Evaluate the limit $\lim_{x \to 2} (x + 3)$. **Step 1:** The function is $f(x) = x + 3$.

1. **Problem:** Evaluate $\lim_{x \to 1^2} x$. **Step 1:** Note that $1^2 = 1$, so the limit is $\lim_{x \to 1} x$.

1. The problem is to evaluate the integral $$\int e^x \sin(x) \, dx$$. 2. We use the method of integration by parts or recognize this as a standard integral involving the product o

1. Problem: Differentiate the function $$f(x) = \ln(x^2 + 1) + 2x$$. 2. Formula: To differentiate a sum, differentiate each term separately. For $$\ln(u)$$, the derivative is $$\fr

1. The problem is to find the indefinite integral $$\int x^2 \sin(2x) \, dx.$$\n\n2. We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du.$$\n\n3. Cho

1. We are asked to find the integral $$\int (r^2 + r + 1)e^r \, dr$$. 2. To solve this, we use integration by parts, which states:

1. **State the problem:** Evaluate the integral $$\int \frac{\sqrt{x}}{x^2+1} \, dx.$$\n\n2. **Rewrite the integrand:** Note that $$\sqrt{x} = x^{\frac{1}{2}}.$$ So the integral be

1. **Stating the problem:** Evaluate the integral $$\int \frac{\sqrt{\sqrt{x}}}{x^2+1} \, dx$$. 2. **Rewrite the integrand:** Note that $$\sqrt{\sqrt{x}} = (x^{1/2})^{1/2} = x^{1/4

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x}{\sqrt{1-3x}}$$. 2. **Recall the formula:** To differentiate a quotient, use the quotient rule: