∫ calculus

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

1. The problem involves understanding and applying double and triple integrals, including changing the order of integration and variables, and using double integrals to find areas