∫ calculus

1. **State the problem:** Find the relative maxima of the function $$y=\frac{1}{3}x^3 + x^2 - 15x + 15$$. 2. **Formula and rules:** To find relative maxima, we first find the criti

1. Pateikime pirmą uždavinį: apskaičiuokime integralą $$\int \sqrt{2 + 3x} \, dx$$. 2. Naudosime pakeitimo metodą. Tegul $$u = 2 + 3x$$, tada $$du = 3 \, dx$$ arba $$dx = \frac{du}

1. **State the problem:** We have a circle defined by the equation $$x^2 + y^2 = 25$$ and the slope of the tangent line at any point on the circle is given by $$\frac{dy}{dx} = -\f

1. **State the problem:** Given the implicit equation $$x^2 z^4 + y^3 z^5 = x + y,$$ find the partial derivatives $$z_x = \frac{\partial z}{\partial x}$$ and $$y z_y = y \frac{\par

1. **State the problem:** Find all relative maxima and minima of the function $g$ given its derivative $g'(x)$. 2. **Recall the critical points:** Critical points occur where $g'(x

1. **State the problem:** We are given three functions: $$y = f(x) = -x^2 + 6x,$$

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