∫ calculus

1. **Problem statement:** Given that $f$ is decreasing on its domain, determine which of the following functions must be increasing on the same domain: (a) $y = -f(x)$

1. Stating the problem: We want to determine in which interval the function $f(x) = 3 - 2e^{3x^2}$ is decreasing. 2. First, we find the derivative $f'(x)$ to analyze where $f$ is i

1. **State the problem:** Given the function $f(x) = \frac{x^4 + 1}{x^2}$, determine the intervals where the function is decreasing. 2. **Rewrite the function:** Simplify $f(x)$ by

1. **State the problem:** Integrate the function $$\frac{x^3-2}{x^2+1}$$ using long division. 2. **Perform long division:** Divide the numerator $$x^3-2$$ by the denominator $$x^2+

1. State the problem: We need to find the indefinite integral $$\int \frac{x^3 - 2}{x^2 + 1} \, dx.$$\n\n2. Simplify the integrand: Perform polynomial division because the degree o

1. السؤال: إذا كانت ن(س) = أ(س)^ص = س^3 + 8 هو التكامل غير المحدد للدالة ن(س)، فما قيمة الثابت أ؟ 2. نعلم أن التكامل غير المحدد لدالة مشتقة يعطي دالة مع ثوابت تكامل.

1. **State the problem:** We need to integrate $$\int \frac{x^3 + 1}{x^2 + 7x + 12} \, dx$$ by using polynomial long division. 2. **Perform long division:** Divide the polynomial n

1. The statement says: "As $x$ goes to negative infinity, $f(x)$ goes to negative infinity." 2. This describes the behavior of the function $f(x)$ as $x$ becomes very large in the

1. The problem states that as $x$ approaches negative infinity, $f(x)$ also approaches negative infinity. We want to understand the behavior and overall shape of the function $f(x)

1. The problem asks whether as $x$ goes to negative infinity, $g(x)$ also goes to negative infinity. 2. From the graph description, the curve starts in the bottom left quadrant goi

1. **Problem:** Evaluate the integral $$\int (1 - 2x)^3 \, dx$$ Step 1: Use substitution. Let $$u = 1 - 2x$$ then $$du = -2 \, dx$$ or $$dx = -\frac{du}{2}$$.

1. We want to find the limit\n\nb) \lim_{x \to 0} \frac{\sin 3x \cdot \tan 4x}{2x^2}\n\n2. Recall the standard limits: \lim_{x \to 0} \frac{\sin ax}{ax} = 1 and \lim_{x \to 0} \fra

1. Differentiate $x^2 \cot x$ using the product rule: $$\frac{d}{dx}(x^2 \cot x) = 2x \cot x + x^2(-\csc^2 x) = 2x \cot x - x^2 \csc^2 x$$

1. **State the problem:** Evaluate the triple integral $$\int_0^\pi \int_0^{\frac{\pi}{2}} \int_0^1 \left(z \sin x + z \cos y\right) \, dz \, dy \, dx.$$\n\n2. **Separate the integ

1. Problem: Integrate the polynomial function $$4x^3 + 6x^2 + 2x + 1$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, w

1. The concept of limits deals with understanding the behavior of a function $f(x)$ as the input $x$ approaches a certain value $a$. 2. Formally, we say the limit of $f(x)$ as $x$

1. Stating the problem: Find the derivative of the function $$y = (5 - 2x)^{-3} + \frac{1}{8} \left(\frac{2}{x} + 1\right)^4.$$\n\n2. Differentiate the first term using the chain r

1. Stating the problem: Find the derivative of the function $$y = (5 - 2x)^{-3} + \frac{8}{1} (x^2 + 1)^4 = (5 - 2x)^{-3} + 8(x^2 + 1)^4$$. 2. Rewrite the function for clarity: $$y

1. **State the problem**: Find the derivative of the function: $$y=\frac{1}{18}(3x-2)^6 + \left(4-\frac{1}{2}x^2\right)^{-1}$$

1. We need to find the derivative of the function $$y=\frac{1}{x}\sin^2(x) - 5x - \frac{x}{3}\cos^3(x).$$ 2. First, rewrite the function clearly as $$y=\frac{\sin^2(x)}{x} - 5x - \

1. **State the problem:** We want to find the derivative of the function $$y=\frac{1}{x}\sin^5 x - \frac{x}{3}\cos^3 x$$. 2. **Rewrite the function for clarity:** $$y=x^{-1}\sin^5