∫ calculus

1. Diberikan integral \( \int_1^3 (2x + 3) \, dx \). Kita diminta mencari nilai integral tersebut. 2. Rumus integral dasar yang digunakan adalah \( \int (ax + b) \, dx = \frac{a}{2

1. The problem asks to find the value of the definite integral $$\int_0^2 (x^2 - 2x + 1) \, dx$$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is:

1. Diberikan integral $$\int_0^{\frac{\pi}{2}} \sin x \, dx$$. Kita diminta mencari nilai integral tersebut. 2. Rumus dasar integral fungsi sinus adalah $$\int \sin x \, dx = -\cos

1. Diberikan integral $$\int_0^{\frac{\pi}{3}} \cos x \, dx$$. Kita diminta mencari nilai integral tersebut. 2. Rumus dasar integral fungsi kosinus adalah $$\int \cos x \, dx = \si

1. The problem states that the gradient (derivative) of the curve at any point $(x,y)$ is given by $2x - 4$. 2. We need to find the equation of the curve $y=f(x)$ such that its der

1. We are asked to find the area between the curve and the x-axis on the interval $[0,b]$ using definite integrals. 2. The formula for the area under a curve $y=f(x)$ from $x=a$ to

1. **State the problem:** Find the area under the curve $y=3x^2$ from $x=0$ to $x=b$ using a definite integral. 2. **Formula:** The area $A$ under a curve $y=f(x)$ from $x=a$ to $x

1. **State the problem:** We analyze the curve given by the function $$y = \frac{x^2 - 49}{x^2 + 5x - 14}$$ to find its domain, derivatives, intervals of increase/decrease, concavi

1. **State the problem:** We want to simplify the function $$y = \frac{x^2 - 49}{x^2 + 5x - 14}$$ to differentiate it, considering the domain restrictions where the denominator is

1. **State the problem:** We need to find the partial derivative $\frac{\partial z}{\partial y}$ from the implicit equation $$x^2 - 3yz^2 + xyz - 2 = 0.$$\n\n2. **Recall the formul

1. **State the problem:** We are given the implicit function $$x^2 - 3yx^2 + xyz - 2 = 0$$ and asked to find the partial derivative $$\frac{\partial z}{\partial y}$$. 2. **Recall t

1. **State the problem:** Find the acute angle between the curves $y_1 = x^2$ and $y_2 = 2x + 3$ at their point of intersection where $x = 1$. 2. **Find the point of intersection:*

1. **Problem:** Use the Intermediate Value Theorem (IVT) to show that the function $$f(x) = e^{x/4} + \frac{1}{8}x^2 - 4$$

1. **State the problem:** Evaluate the integral $$\int \frac{\sin 3x}{1 + \cos 3x} \, dx.$$\n\n2. **Recall the formula and rules:** We can use the substitution method and trigonome

1. Problem: Find the first partial derivatives of $$z = x^3 - 3x^2 y^4 + y^2$$ with respect to $$x$$ and $$y$$. 2. Formula: The partial derivative of $$z$$ with respect to $$x$$ is

1. **Problem Statement:** Approximate the integral $$\int_0^{0.35} \frac{2}{x^2 - 4} \, dx$$ using three numerical methods: (a) Gaussian quadrature with $n=3$, (b) Closed Newton-Co

1. **Problem statement:** Evaluate the integral $$\int 2x(x^2 + 4)^3 \, dx$$ using the substitution $$u = x^2 + 4$$. 2. **Formula and substitution:** We use substitution for integr

1. **Problem:** Find the volume of the solid of revolution formed by revolving the region bounded by $y = x^2$ and $x = y^2$ about the line $y = 1$. 2. **Identify the curves and re

1. The problem asks to select the correct options for given integration and calculus questions. 2. (i) If $f$ is integrable, it must be continuous almost everywhere, so the best ch

1. The problem asks to identify the function $f$ and the number $a$ for which the limit $$\lim_{h \to 0} \frac{\sqrt[5]{32 + h} - \sqrt[5]{32}}{h}$$

1. **State the problem:** Calculate the limit $$\lim_{x \to 1} \frac{\sqrt{3x - 1} - \sqrt{3 - x}}{x - 1}$$