∫ calculus

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ given $y = u^5 - 2u^3 + 8$ and $u = x^2 + 1$. 2. **Formula and rules:** Use the chain rule for derivatives:

1. **State the problem:** We want to approximate the length of the graph of the function $f$ on the interval $[1,7]$ using a left Riemann sum with 3 subintervals of equal length. 2

1. **State the problem:** Calculate the definite integral $$F(10) - F(0) = \int_0^{10} 20te^{-0.8t} \, dt$$ using integration by parts. 2. **Recall the integration by parts formula

1. **Stating the problem:** Calculate the derivative of the function $f(x) = x^2 - 2x$ at the point $x_0 = 2$ using the definition of the derivative. 2. **Definition of derivative:

1. **Stating the problem:** We want to understand and apply Green's Theorem, which relates a line integral around a simple closed curve $C$ to a double integral over the plane regi

1. The problem is to find the integral of $\sin \theta$ with respect to $\theta$. 2. The formula for the integral of $\sin x$ is:

1. Calcola la derivata prima di $f(x) = \sqrt{\sin^2 x + \cos^2 x}$. La funzione sotto radice è $\sin^2 x + \cos^2 x$, che è sempre uguale a 1 per l'identità trigonometrica fondame

1. **State the problem:** We need to find the slope of the tangent line to the curve at the point $(-1,0)$. The slope of the tangent line represents the derivative of the function

1. **State the problem:** Find the area bounded by the polar curve given by $$r = 4 - 3\cos\theta$$. 2. **Formula for area in polar coordinates:** The area enclosed by a polar curv

1. **State the problem:** Find the coordinates of the centroid $(\bar{x}, \bar{y})$ of the shaded region bounded by the curves $f_1(x) = (x-4)^2$ and $f_2(x) = 4$ between $x=2$ and

1. **State the problem:** Evaluate the integral $$\int \cos x \sqrt{\sin x} \, dx$$ using substitution. 2. **Identify substitution:** Let $$u = \sin x$$, then $$du = \cos x \, dx$$

1. **State the problem:** Evaluate the definite integral $$\int_1^4 \frac{4x+1}{2x^2+x} \, dx$$. 2. **Simplify the integrand:** Factor the denominator:

1. **State the problem:** We need to evaluate the definite integral $$\int_{-2}^{3} (4x^2 + 2x - 5) \, dx$$. 2. **Recall the formula:** The integral of a polynomial $$ax^n$$ is $$\

1. **State the problem:** We want to find the dimensions of an open rectangular box with a square base that maximizes volume, given that the surface area is 48 ft\(^2\). 2. **Defin

1. The problem involves finding the area between two curves using integral calculus. 2. The general formula for the area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b

1. **State the problem:** Find the point on the curve $y=\sqrt{x}$ where the tangent line is parallel to the line $y=\frac{1}{8}x$. 2. **Identify the slope of the given line:** The

1. **State the problem:** Find the limit \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \). 2. **Recall the formula and rules:** The limit of a rational function as \( x \) approaches a v

1. **State the problem:** Evaluate the definite integral $$\int_{\sqrt{2}}^{2} \frac{1}{t^3 \sqrt{t^2 - 1}} \, dt.$$\n\n2. **Identify the integral type and substitution:** The inte

1. **Problem:** Bestäm värdet av integralen $$\int_4^{12} f(x) \, dx$$ från grafen av funktionen $y = f(x)$. Endast svar krävs. 2. **Problem:** Avgör om integralen $$\int_a^b f(x)

1. **State the problem:** Evaluate the integral $$\int 4 \sin x \cos x \, dx$$. 2. **Recall the formula:** Use the double-angle identity for sine: $$\sin(2x) = 2 \sin x \cos x$$.

1. **State the problem:** We need to evaluate the integral $$\int 2x \sin x \, dx$$. 2. **Formula and rules:** Use integration by parts, which states: