∫ calculus

1. The problem is to integrate the function given as "the answer". Since no specific function is provided, let's assume the problem is to find the indefinite integral of a general

1. Evaluate the limits using limit laws: (i) Find $$\lim_{x \to 2} \frac{2x^3 - 3x + 1}{x^3 + 4}$$

1. **State the problem:** Find the exact value of the integral $$\int_0^{\frac{\pi}{6}} x^2 \sin(2x) \, dx$$. 2. **Formula and method:** We will use integration by parts, which sta

1. **State the problem:** We want to evaluate the double integral

1. We are asked to find the exact value of the integral $$\int_0^1 x \tan^{-1}(x) \, dx.$$\n\n2. To solve this, we use integration by parts. Recall the formula: $$\int u \, dv = uv

1. **State the problem:** Find the exact value of the integral $$\int_1^2 \ln(3x) \, dx$$ in the form $a + \ln b$, where $a$ and $b$ are integers. 2. **Recall the integration formu

1. The problem is to find the derivative of the denominator if the denominator is $1 + t$. 2. The derivative of a function $f(t)$ with respect to $t$ is denoted as $\frac{d}{dt}f(t

1. **State the problem:** We want to find the integral $$L(t) = \int \frac{1+t}{t^2 + 3t} \, dt.$$ 2. **Rewrite the integrand:** Factor the denominator: $$t^2 + 3t = t(t+3).$$ So t

1. **Problem statement:** Express the function $$f(x) = \frac{12 + 8x - x^2}{(2 - x)(4 + x^2)}$$ in the form $$\frac{A}{2 - x} + \frac{Bx + C}{4 + x^2}$$ and then show that $$\int_

1. The problem is to find the integral $$\int \tan^2(5x) \, dx$$. 2. Recall the identity $$\tan^2(\theta) = \sec^2(\theta) - 1$$. This helps us rewrite the integral in terms of sec

1. **Problem Statement:** We are given the graph of the derivative $f'$ of a function $f$ on the interval $[-9,9]$ and the value $f(9) = -2$. We need to find the maximum value of $

1. Stwierdźmy problem: obliczyć całkę \( \int \ln(2x+1) \, dx \). 2. Użyjemy metody całkowania przez części, gdzie \( u = \ln(2x+1) \) i \( dv = dx \).

1. **State the problem:** Evaluate the integral $$\int \frac{\sec\left(\frac{1}{v}\right) \tan\left(\frac{1}{v}\right)}{v^2} \, dv.$$\n\n2. **Identify substitution:** Let $$u = \fr

1. **Problem:** Find $\lim_{x \to 0} \frac{\sin(5x)}{2x}$.\n\n2. **Formula and rules:** We use the standard limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$.\n\n3. **Rewrite the limit:*

1. **Problem:** Find $$\lim_{x \to 0} \frac{\sin(5x)}{2x}$$. 2. **Formula and rules:** We use the standard limit $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. **State the problem:** Find the area bounded by the curve $y = x^2 - 2x$, the $x$-axis, and the vertical lines (ordinates) $x = -2$ and $x = 3$. 2. **Formula and rules:** The ar

1. **State the problem:** Find the derivative of the function $$y = -8x^{-8} + 12\sqrt{x}$$ using the power rule. 2. **Recall the power rule:** For any function $$y = x^n$$, the de

1. **State the problem:** We need to determine the integral of the function $$\frac{x^3+5}{x^2-25}$$ with respect to $x$. 2. **Recall the formula and rules:** To integrate a ration

1. **State the problem:** We need to determine the derivative of the function $$f(x) = \frac{x^3 + 5}{x^2} - 25$$ with respect to $$x$$. 2. **Rewrite the function:** To differentia

1. **Problem statement:** Find the indefinite integral $$\int \frac{1}{t^4} \, dt$$. 2. **Rewrite the integrand:** Recall that $$\frac{1}{t^4} = t^{-4}$$.

1. **Problem statement:** (a) Given that $|f(x) - 3| \leq 4(x - 2)^2$, find $\lim_{x \to 2} f(x)$.