∫ calculus

1. **State the problem:** We need to evaluate the integral $$\int 7x \sqrt{1 - x^4} \, dx$$. 2. **Identify a substitution:** Notice the expression inside the square root is $1 - x^

1. **State the problem:** We need to evaluate the integral $$\int \frac{10}{x \sqrt{x^2 - 36}} \, dx$$. 2. **Recall the formula and substitution:** For integrals involving expressi

1. **State the problem:** We need to evaluate the integral $$\int \frac{-10}{x \sqrt{x^2 - 1}} \, dx$$. 2. **Recall the formula and substitution:** For integrals involving expressi

1. نبدأ ببيان المشكلة: لدينا دالة معرفة كالتالي: $$f(x)=\begin{cases} x^2 & x \neq 1 \\ 2 & x=1 \end{cases}$$

1. **Problem statement:** We need to find the range of values of $a$, with $0 \leq a \leq 5$, for which the integral $$\int_0^a f(x) \, dx$$ is positive. 2. **Understanding the gra

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$$ algebraically. 2. **Recall the formula and rules:** To find limits involving rational expression

1. **State the problem:** Find the limit \(\lim_{n \to 0} \frac{(4+n)^2 - 16}{n}\). 2. **Recall the formula:** This is a difference quotient form, often used to find derivatives.

1. **State the problem:** Find the derivative of the function $$y = \frac{x^3}{4} - 3x$$. 2. **Recall the derivative rules:**

1. The problem involves understanding the equation with partial derivatives: $$\partial V + \partial U + \partial Q_{totao} = Q_{Ev} = \partial Z_c$$. 2. This equation suggests a r

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^5 - x}{x - 1}$$ using algebraic methods. 2. **Recall the formula and rules:** When direct substitution results in

1. The problem is to solve an integral using integration by parts. 2. The formula for integration by parts is:

1. **Stating the problem:** Evaluate the integral $$\int (\cos^{-1} 3x) \frac{x}{\sqrt{1 - 9x^2}} \, dx.$$\n\n2. **Understanding the integral:** The integrand is a product of the i

1. **State the problem:** Find the limits \( \lim_{x \to 9} \frac{3 + \sqrt{x}}{x - 9} \) and \( \lim_{t \to -1} \frac{t^2 - 3t - 4}{t^3 - 1} \). 2. **First limit:** \( \lim_{x \to

1. **State the problem:** We need to evaluate the definite integral $$\int_{-2}^1 x \sqrt{3+x} \, dx$$. 2. **Substitution:** Let $$u = 3 + x$$, then $$du = dx$$ and when $$x = -2$$

1. The problem asks to mark all the relative minimum points on the given graph. 2. A relative minimum point on a graph is a point where the function changes from decreasing to incr

1. **State the problem:** Find the inflection point(s) of the function $$g(x) = \frac{x^2}{2} (9 - 2 \ln(x))$$ and analyze its concavity. 2. **Recall formulas:**

1. The problem is to understand how to handle the limit definition when the limit $L = -\infty$. 2. The standard limit definition for a finite limit $L$ is: for every $\epsilon > 0

1. **State the problem:** Prove using the formal definition of limits that $$\lim_{x \to 0} -\frac{1}{x^2} = -\infty$$. 2. **Recall the formal definition for limits tending to nega

1. **State the problem:** We have two functions $f(x)=(x+5)^2$ and $g(x)=(x-7)^2$. Their tangents have gradients related by $m_f=1.5m_g$. We want to find the equations of the tange

1. We start with the problem: find the definite integral (ondersom) of the function $f(x) = x^3$ given that the integral of $f(x) = x^2$ is $\frac{1}{3}$. 2. Recall the formula for

1. **Problem statement:** Calculate the antiderivative (indefinite integral) of the function $$\int \frac{x}{\sqrt{1-x^4}} \, dx$$. 2. **Formula and approach:** To solve integrals