∫ calculus

1. **State the problem:** We are given the function $s = \sqrt{t} + 2$ and need to find the derivative $\frac{ds}{dt}$ at $t = 7$. 2. **Recall the formula:** The derivative of $s$

1. Mari kita selesaikan soal pertama bagian a: Derivasi fungsi $Y = (2X + 5)(4X^2)$. 2. Gunakan aturan perkalian untuk turunan: Jika $Y = u \cdot v$, maka $Y' = u'v + uv'$.

1. **Problem 1: Differentiate $y = x \sin x$ with respect to $x$.** 2. Use the product rule: If $y = uv$, then $\frac{dy}{dx} = u'v + uv'$. Here, $u = x$, $v = \sin x$.

1. **Problem:** Evaluate the integral $$\int 3x^5 \sqrt{16 - x^2} \, dx$$. 2. **Formula and rules:** We will use substitution and algebraic manipulation. Recall that substitution i

1. **Problem 1a:** Find the derivative of $y = 5x^5$. Formula: For $y = ax^n$, the derivative is $y' = a n x^{n-1}$.

1a. Differentiate $y=5x^5$ using the power rule $\frac{d}{dx}x^n = nx^{n-1}$. $$\frac{dy}{dx} = 5 \times 5x^{5-1} = 25x^4$$

1. The problem is to evaluate the improper integral $$\int_0^\infty \frac{x}{(1+x^2)^2} \, dx$$ over the infinite interval from 0 to infinity. 2. We use the formula for integration

1a. Given $y = 5x^5$, find the derivative $\frac{dy}{dx}$. Using the power rule $\frac{d}{dx} x^n = nx^{n-1}$, we get

1. **Problem Statement:** Calculate the definite integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\tan^{-1}(x)\right)^2 \, dx$$. 2. **Formula and Rules:** The function involv

1. **Problem Statement:** Find the area of the region enclosed between a given curve and a line. 2. **General Approach:** To find the area between a curve $y=f(x)$ and a line $y=g(

1. Differentiate $y = x \sin x$ with respect to $x$. Use the product rule: $\frac{d}{dx}[uv] = u'v + uv'$.

1. **Problem:** Solve the integral that leads to the expression $\ln(|\ln|x||) + c$. 2. **Formula and rules:** The integral involves nested logarithms. Recall that $\frac{d}{dx} \l

1. **State the problem:** We are given that $f$ is continuous and $$\int_0^9 f(x) \, dx = 10,$$

1. We are asked to evaluate the integral $$\int \frac{e^{\sqrt{6y + 4}}}{\sqrt{6y + 4}} \, dy.$$\n\n2. To solve this, use the substitution method. Let $$u = \sqrt{6y + 4}.$$\n\n3.

1. Problem statement: Compute the indefinite integrals in 4(a), evaluate the indefinite integrals in 4(b), solve the antiderivative and area problems in 5, and carry out the differ

1. **Stating the problem:** We are given the volume change rate of a balloon as $\frac{dV}{dt} = 1.08\pi$ cm³/s and asked to find the rate of change of the radius $\frac{dr}{dt}$ w

1. **Problem Statement:** Set up the integral to calculate the volume of the region bounded by the cylinder $z = y^2$, the planes $x=0$, $x=1$, $y=-1$, $y=1$, and the XY-plane ($z=

1. The problem is to differentiate the function, but the function itself was not provided. 2. To differentiate a function $f(x)$, we use the derivative rules such as the power rule

1. **State the problem:** Differentiate the function $$y = 2x \log_3(\sqrt{x})$$. 2. **Recall the formula and rules:**

1. **State the problem:** Find the derivative of the function $$y = x^{3x}$$ using logarithmic differentiation. 2. **Recall the formula and rules:** For functions of the form $$y =

1. **State the problem:** We are given the derivative of a function $f'(x) = e^{-x^2} (x^2 - 1)(2x - 3)$ and asked to find all values of $x$ where the tangent line to $f(x)$ is hor