∫ calculus

1. **Problem statement:** Calculate the integral $$\int \sqrt{x} (x + 2) \, dx$$. 2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the integrand becomes

1. **State the problem:** We need to verify and fill in the blanks for the integral $$\int \frac{16x - 24x^2}{x^4} \, dx = -16 x^{-2} + 24 x^{-1} + C$$

1. **State the problem:** We want to approximate the distance traveled by a particle over the time interval $0 \leq t \leq 8$ seconds using the midpoint Riemann sum with four subin

1. **State the problem:** Determine if the infinite series $$\sum_{n=1}^{\infty} \frac{n^2 - 1}{2n^3 + 1}$$ converges using the $n^{th}$ term test for convergence. 2. **Recall the

1. The problem asks for the least number of terms needed to approximate the alternating series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}\) within an error of \(\pm 0.001\)

1. **State the problem:** Evaluate the integral $$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2\left(\frac{\pi}{2}x\right)}{\left(x - \frac{\pi}{2}\right)^2} \, dx.$$ 2. **Analyze the inte

1. **State the problem:** We need to find the area of the region bounded by the x-axis, y-axis, and the curve $$\sqrt{x} + \sqrt{y} = 3$$. 2. **Rewrite the curve equation:** Solve

1. **State the problem:** We need to find the indefinite integral $$\int 15\sqrt{4 + x} \, dx$$ and express the answer in the form $$a(4 + x)^b + C$$ where $a$ and $b$ are constant

1. **State the problem:** We need to find the values of $a$, $b$, $c$, $p$, $q$, $s$, and $r$ in the expression for the area $S$ of the shaded region bounded by the curves $y=f(x)=

1. **State the problem:** Evaluate the definite integral $$\int_7^8 x\sqrt{x} - 7 \, dx$$ with four decimal places. 2. **Rewrite the integrand:** Note that $$\sqrt{x} = x^{\frac{1}

1. **State the problem:** Evaluate the integral $$\int \frac{1 + 41x}{1 + x^2} \, dx.$$\n\n2. **Recall the formula and rules:** We can split the integral into two simpler integrals

1. **State the problem:** We need to estimate the gradient (slope) of the curve $y=f(x)$ at the point where $x=3$. 2. **Understanding the gradient:** The gradient at a point on a c

1. **Énoncé du problème :** Corriger l'exercice 3 qui porte sur la fonction $g(x) = x \sqrt{x + n}$ avec $\sqrt{x} \in [-1, +\infty[$.

1. **Сформулюємо задачу:** Обчислити визначені та невласні інтеграли: а) $$\int_{-2}^{-1} (x+2)^2 \cos 3x \, dx$$

1. The problem states the operator $$H(r) = \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial r} (r h) \right)$$ where $$r = \sqrt{x^2 + y^2}$$. 2. This opera

1. **State the problem:** Find the limit as $x$ approaches infinity of $$\frac{-4x^2}{\sqrt{16x^4 + 3}}.$$\n\n2. **Recall the formula and rules:** When dealing with limits involvin

1. **State the problem:** We have a piecewise function $$f(x) = \begin{cases} 4x^2 + ax - 1 & \text{if } 0 \leq x < 2 \\ 3x + a & \text{if } x > 2 \end{cases}$$

1. Állítsuk fel a problémát: Számítsuk ki az integrált $$\int_2^3 \frac{6}{x^2} \, dx$$. 2. Használjuk az integrálási szabályt: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, ahol $

1. **State the problem:** Calculate the definite integral $$\int_1^6 x \, dx$$. 2. **Formula and rules:** The integral of $$x$$ with respect to $$x$$ is given by $$\int x \, dx = \

1. **State the problem:** Calculate the limit $$\lim_{x\to +\infty} \frac{1 - 6x^3}{2x^3 - x^2 + 5x - 3} + \lim_{x \to -\infty} \frac{1}{|x| + 1}$$

1. **Problem:** Find the derivative of the function $f(x) = \frac{8}{x} - x$. 2. **Formula:** The derivative of a function $f(x)$ is given by $f'(x) = \lim_{h \to 0} \frac{f(x+h) -