∫ calculus

1. The problem asks to find the derivative with respect to $x$ of the function $a^3 \log_a x$. 2. Recall the formula for the derivative of $\log_a x$:

1. **State the problem:** We need to evaluate the integral $$\int 15x \sqrt{x} + 4 \, dx$$. 2. **Rewrite the integral:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the integral

1. **Problem:** Evaluate the limit $$\lim_{t \to 0} \frac{\sin t}{t}$$ using special limits. 2. **Formula and important rule:** The special limit $$\lim_{x \to 0} \frac{\sin x}{x}

1. The problem is to find the indefinite integral of the function $$f(x) = \frac{2}{\sqrt{x}} + 6x - 3 \cdot 2^{4x} + \left(\frac{1}{3}\right)^{-x}$$ with respect to $x$. 2. Recall

1. **State the problem:** We want to find the limit as $x$ approaches 0 of the expression $$\frac{e^{3x} - 1 - 3x}{x^2}.$$\n\n2. **Recall the formula and rules:** The exponential f

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} \frac{e^x - \cos x - x}{x^2}$$ using series expansions. 2. **Recall the series expansions:**

1. **State the problem:** Find the derivative of the function $f(x) = 3x^2 - 11$ at $x = 2$ using the limit definition of the derivative. 2. **Recall the limit definition of the de

1. The problem is to evaluate the integral $$\int \sin x \, dx$$. 2. The formula for the integral of sine is $$\int \sin x \, dx = -\cos x + C$$, where $C$ is the constant of integ

1. **Énoncé du problème :** Calculer l'intégrale $$\int_0^1 x e^x \, dx$$ en utilisant l'intégration par parties. 2. **Formule d'intégration par parties :** $$\int u \, dv = uv - \

1. **State the problem:** We need to evaluate the double integral $$\int_{1}^{3} \int_{-2}^{4} (x^2 - 2xy + 3y^2 - 5) \, dy \, dx$$. 2. **Integrate with respect to $y$ first:** Tre

1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = -4 \sin(x) + 9x.$$\n\n2. **Recall the derivative rules:**\n- The derivative of $\sin(x)$

1. The problem asks to estimate the derivative $f'(0)$ of the function $f(x)$ at $x=0$. 2. The derivative $f'(x)$ at a point gives the slope of the tangent line to the graph of $f(

1. **Stating the problem:** Find the general antiderivative $F(x) + C$ of the function $f(x) = 2x - 4$. 2. **Formula used:** The antiderivative (indefinite integral) of a function

1. Diberikan fungsi $f(x) = 4$. Kita diminta mencari anti turunan umum $F(x) + C$. 2. Rumus anti turunan dasar adalah $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ dan

1. **Problem statement:** Find the volume generated by revolving the curve $f(x) = 4 - x^2$ bounded by $x_1 = -2$ and $x_2 = 2$ around the x-axis.

1. **State the problem:** Evaluate the integral $$\int \frac{2}{3x^2} \, dx$$. 2. **Rewrite the integral:** We can rewrite the integrand as $$\frac{2}{3x^2} = \frac{2}{3} x^{-2}$$.

1. The problem is to find the derivative of a function using basic derivative rules. 2. The basic derivative rules include:

1. Evaluate limits for $g(x) = \frac{-4}{(x-1)^2}$: (a) As $x \to 1^-$, $(x-1)^2$ approaches 0 from the positive side since squaring any real number is positive.

1. Soal: Tentukan turunan kedua dari fungsi $f(x) = x^2 + x!$ menggunakan definisi turunan. 2. Definisi turunan pertama pada titik $c$ adalah:

1. The user asked for questions on Calculus III, which typically involves multivariable calculus topics such as partial derivatives, multiple integrals, and vector calculus. 2. Sin

1. Evaluate limits for $g(x) = \frac{-4}{(x-1)^2}$: (a) $\lim_{x \to 1^-} g(x)$: As $x$ approaches 1 from the left, $(x-1)^2$ approaches 0 from the positive side (since square is a