∫ calculus

1. **Problem 5:** Evaluate $$\frac{d}{dx} \int_1^{5x} e^{t^2} dt + \int_1^x \frac{d}{dx} \left[ \sqrt{1 + \ln(x)} \right] dx + \frac{d}{dx} \int_{\pi}^{2\pi} \frac{\sin x}{x} dx.$$

1. **State the problem:** Find the integral of the function $2x^2$ with respect to $x$. 2. **Recall the formula:** The integral of $x^n$ with respect to $x$ is given by

1. **Problem statement:** Calculate the volume of the solid formed by rotating the function $f(x) = 2x^2 + 5x + 2$ over the interval $[0,4]$ around the x-axis.

1. **State the problem:** We want to use the substitution $x = u^2 + 1$ to show that

1. **State the problem:** Evaluate the definite integral $$\int_0^2 2x \sqrt{x+2} \, dx$$ and verify that it equals $$\frac{32}{15}(2 + \sqrt{2})$$. 2. **Set up the integral:** We

1. **State the problem:** We need to find the equation of the tangent line to the curve defined by the function $$f(x) = 2x^3 - 9x^2 + 5x - 11$$ at the point where $$x = 2$$. 2. **

1. **Problem statement:** Given the function $F(t) = -50t e^{-0.2t} - 250 e^{-0.2t}$, find the function $f(t)$ and its first derivative $f'(t)$. 2. **Rewrite the function:** Notice

1. **Stating the problem:** We want to analyze the behavior of the function on the interval $6 < x < 7$ based on the graph description. 2. **Understanding the graph:** The function

1. **State the problem:** Find the integral of $$\left(\frac{x}{x-2}\right)^2$$ with respect to $x$. 2. **Rewrite the integrand:** Simplify the expression inside the integral.

1. **State the problem:** We need to find the integral of the function $f(x) = (1 - x) \sqrt{2x - 3}$. That is, compute $$\int (1 - x) \sqrt{2x - 3} \, dx.$$\n\n2. **Rewrite the in

1. **Problem statement:** Given the derivative function $$f'(x) = \frac{-5x}{(x^2+1)^2}$$, analyze its behavior and find critical points. 2. **Recall the formula:** Critical points

1. **State the problem:** Find the integral $$\int \frac{x^2}{\sqrt{x-1}} \, dx$$. 2. **Rewrite the integral:** Let us use substitution to simplify the integral. Set $$u = x - 1$$,

1. **State the problem:** Evaluate the integral $$\int x^2 \sqrt{1-x} \, dx$$ using substitution. 2. **Choose substitution:** Let $$u = 1 - x$$. Then, $$du = -dx$$ or $$dx = -du$$.

1. **State the problem:** We want to evaluate the integral $$\int 2x \sqrt{x-1} \, dx$$ using substitution. 2. **Choose a substitution:** Let $$u = x - 1$$. Then, $$du = dx$$ and $

1. **State the problem:** We need to find the integral $$\int 2x \sqrt{x-1} \, dx$$ using substitution. 2. **Choose substitution:** Let $$u = x - 1$$. Then, $$du = dx$$ and $$x = u

1. Énoncé du problème : Nous avons la fonction $z(x,y) = 2 + xy + \frac{x^2}{4}$ avec $x \in [0,2]$ et $y \in [0,2]$. Nous devons estimer l'aire de la surface définie par cette fon

1. We are asked to find the integral $\int \sin 5x \, dx$. 2. The formula for integrating sine functions with a linear argument is:

1. **Problem:** Evaluate the integral $$\int \sqrt{x - 2} \, dx$$ 2. **Formula and substitution:** Use substitution $u = x - 2$, so $du = dx$. The integral becomes $$\int u^{1/2} \

1. **Stating the problem:** We are given the derivatives of three functions: $$f'(x) = -3x + 4$$

1. **State the problem:** Find the derivative of the function $y = \sin(\cot(5x))$ with respect to $x$. 2. **Recall the chain rule:** If $y = \sin(u)$ where $u = \cot(5x)$, then by

1. We are asked to evaluate the definite integral $$\int_0^4 (2e^x + 3 \sin x) \, dx$$. 2. The integral of a sum is the sum of the integrals, so we split it: