∫ calculus

1. **Stating the problem:** Evaluate the double integral

1. **State the problem:** We want to find the integral $$\int \sqrt{\cos x (3 + 2 \cos x)} \, dx$$. 2. **Rewrite the integrand:** The expression inside the square root is $$\cos x

1. **State the problem:** We need to evaluate the integral $$\int \frac{dx}{\sqrt{\cos x (3 + 2 \cos x)}}.$$\n\n2. **Rewrite the integral:** The integral is $$\int \frac{dx}{\sqrt{

1. **Problem I:** Evaluate $$\int \frac{dx}{x \sqrt{x^2 - \pi}}$$ - Use substitution for integrals involving $$\sqrt{x^2 - a^2}$$.

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ for the equation $$e^{x+y} = \cos\left(e^y\right) + \ln(xy).$$ 2. **Recall the rules:** We will use implicit different

1. **State the problem:** We are given the double integral

1. **State the problem:** Given the function $$y(x) = \frac{12x^6 - 4x^4 + 3x^2 - 1}{8x^4}$$, find its derivative $$y'(x)$$ and verify the given derivative $$y'(x) = 3x - \frac{3}{

1. **State the problem:** Find the derivative of the function $$y(x) = \frac{3x^2 - 2x + 2}{-5x^2 + 3x - 1}$$.

1. **Problem:** Find local and global extrema of $f(x) = 4 - x^2$ on the interval $[-3, 1]$. 2. **Formula and rules:** To find extrema, first find critical points by setting the de

1. **State the problem:** Differentiate the function $$y(x) = \frac{x^3 + 1}{5 - 2x^3}$$ with respect to $x$. 2. **Recall the quotient rule:** For a function $$y = \frac{u(x)}{v(x)

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^3 - 3x + 2}{x^3 - 4x + 3}$$. 2. **Check direct substitution:** Substitute $x=1$ into numerator and denominator:

1. **Limits of Algebraic Functions:** - $\lim_{x \to a} c = c$

1. **Problem:** Find the value of the function $f(x) = 3x^2 - 5x + 2$ at $x=4$. 2. **Problem:** Evaluate the limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.

1. Problem: Find the value of the limit $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$. 2. Formula: The limit of a function as $x$ approaches a value $a$ is the value that $f(x)$ approac

1. **State the problem:** We need to find \(\frac{dy}{dx}\) for the equation $$2 y^3 + 2 x^2 = 4$$ using implicit differentiation. 2. **Recall the rule:** When differentiating impl

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{x^2 - 2x + 7}{2x^2 - 3x - 4}$$ using limit laws. 2. **Recall limit laws:** If the limit of numerator and denomin

1. **State the problem:** We need to find the intervals where the function $$f(x) = x - \frac{1}{6}x^2 - \frac{1}{3}\ln x + \frac{1}{6}$$ is concave up and concave down, and also f

1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} n\left(\sqrt{2n^2+1} - \sqrt{2n^2-1}\right).$$ 2. **Use the conjugate to simplify:** To simplify the expre

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ if $y = (\cos x)^{\sin x}$ using logarithmic differentiation. 2. **Take the natural logarithm of both sides:**

1. **State the problem:** Find the derivative of the function $f(x) = \sin^2(x) \times \tan(x)$. 2. **Recall the formula:** To differentiate a product of two functions $u(x)$ and $

1. **State the problem:** We want to evaluate the integral $$\int \frac{7x^3 - 8x + 5}{x^3 - x^2} \, dx$$ by first performing long division, then expressing the proper fraction as