∫ calculus

1. Diberikan fungsi $f(x) = 2x^{2} + 3x + 5$, kita diminta menghitung turunan pertama pada $x=2$ dengan $h=0,1$ menggunakan tiga metode: beda maju, beda mundur, dan beda pusat. 2.

1. **Problem:** Evaluate the integral $$\int_0^3 \sqrt{y+1} \, dy$$. 2. **Formula and rules:** Use the substitution for integrals of the form $$\int \sqrt{u} \, du = \int u^{1/2} \

1. The problem asks about the meaning of the statement: "Odd function over symmetric limit is zero." 2. An odd function $f(x)$ satisfies the property $f(-x) = -f(x)$ for all $x$ in

1. The problem asks to evaluate the definite integral $$\int_{-a}^a (x^3 + x) \, dx$$. 2. Recall that the integral of a sum is the sum of the integrals:

1. Soal pertama: Hitung nilai turunan pertama $f(x) = 2x^2 + 3x + 5$ pada $x=2$ dengan $h=0,1$ menggunakan metode hampiran beda maju, beda mundur, dan beda pusat. 2. Rumus turunan

1. **Problem statement:** Show that the function satisfies the hypotheses of Rolle's Theorem on the interval [0,4] and find all numbers $c$ in $(0,4)$ such that $f'(c) = 0$ for $f(

1. **State the problem:** Solve the differential equation $$xy'' = y' \ln\left(\frac{y'}{x}\right)$$. 2. **Rewrite the equation:** Let $p = y' = \frac{dy}{dx}$ and $p' = y'' = \fra

1. The problem is to understand and use partial derivatives. 2. Partial derivatives measure how a multivariable function changes as one variable changes, keeping others constant.

1. The problem is to evaluate the sum of integrals: $$\sum_{k=1}^\infty \int_0^\infty x^3 e^{-kx} \, dx$$

1. **Problem:** Sketch the region of integration for the integral $$\int_1^{10} \int_0^{\frac{1}{y}} y e^{xy} \, dx \, dy$$ and evaluate it. 2. **Understanding the region:** The li

1. **State the problem:** We need to evaluate the integral $$\int x^3 \sqrt{1+x^2} \, dx$$. 2. **Identify a substitution:** Let $$u = 1 + x^2$$. Then, $$du = 2x \, dx$$ or $$x \, d

1. **State the problem:** We need to evaluate the integral $$\int x^3 \sqrt{1x^2} \, dx$$. 2. **Simplify the integrand:** Since $$\sqrt{1x^2} = \sqrt{x^2} = |x|$$, the integral bec

1. The problem is to find the limit of a function as $x$ approaches $+\infty$. 2. The limit notation is $\lim_{x \to +\infty} f(x)$, which means we want to see what value $f(x)$ ap

1. The problem is to find the limit of the function $e^{-x}$ as $x$ approaches infinity. 2. The function is $f(x) = e^{-x}$.

1. **State the problem:** We need to find the integral $$\int \frac{x^2}{(x-2)^{10}} \, dx$$. 2. **Rewrite the integrand:** Let’s use substitution to simplify the integral. Set $$u

1. The problem is to evaluate the integral $$\int \frac{x^2}{x} \, dx$$. 2. Simplify the integrand by dividing $x^2$ by $x$ which gives $x$.

1. The problem is to find the integral of $\sin^2 x$ with respect to $x$. 2. We use the trigonometric identity to simplify the integrand: $$\sin^2 x = \frac{1 - \cos(2x)}{2}$$

1. The problem is to find the indefinite integral of $\cos\left(\frac{x}{3}\right)$ with respect to $x$. 2. Recall the formula for integrating cosine of a linear function: $$\int \

1. **State the problem:** Evaluate the integral $$\int \frac{\cos x}{\sin x} + 3 \, dx$$. 2. **Rewrite the integral:** Split the integral into two parts:

1. Let's start by stating the problem: We want to understand the differentiability of a function of two variables in a simple way. 2. Consider a function $f(x,y)$ that depends on t

1. **State the problem:** We need to evaluate the double integral $$\int_{-2}^2 \int_x^2 \cos(\sqrt{y^3}) \, dy \, dx.$$\n\n2. **Understand the integral:** The outer integral is wi