∫ calculus

1. Problem 1: Evaluate the integral $$\int 12 \sin^6(x) \cos^3(x) \, dx$$ 2. To solve this, we use substitution and trigonometric identities. Notice the powers of sine and cosine.

1. **State the problem:** We need to evaluate the integral $$\int 45 \sin^3(x) \cos^2(x) \, dx.$$\n\n2. **Rewrite the integral:** Use the identity $$\sin^3(x) = \sin(x) \cdot \sin^

1. **State the problem:** We want to evaluate the integral $$\int 5 \tan^3(x) \sec(x) \, dx.$$\n\n2. **Recall relevant formulas and identities:**\n- We know that $\frac{d}{dx}(\sec

1. **State the problem:** We want to solve the integral $$\int 5x \sqrt{1 - x^4} \, dx$$. 2. **Identify a substitution:** Notice the expression inside the square root is $1 - x^4$.

1. **State the problem:** Differentiate the function $f(x) = (6e^{2x} - 3)^3$ with respect to $x$. 2. **Recall the chain rule:** If $y = [u(x)]^n$, then $\frac{dy}{dx} = n[u(x)]^{n

1. The problem is to calculate the limit $$\lim_{n \to \infty} (\sqrt{2n + 1} - \sqrt{n + 2})$$. 2. To solve limits involving differences of square roots, we use the conjugate to s

1. **Stating the problem:** Calculate the limit

1. **State the problem:** Find the limit

1. **Problem statement:** Evaluate the integral $$\int \frac{1}{r} e^{rx} \cos x \, dx$$. 2. **Given substitutions:**

1. مسئله: مساحت سطح حاصل از دوران منحنی $y=f(x)$ حول محور $x$ را در بازه $[a,b]$ بیابید. 2. فرمول: مساحت سطح دوران حول محور $x$ برابر است با:

1. **Problem statement:** Calculate the integral $$\int \frac{\ln x}{x} \, dx$$. 2. **Recall integration by parts formula:**

1. **State the problem:** We are given the second derivative of a function $f''(x) = x^{-3/2}$, with initial conditions $f'(4) = 2$ and $f(0) = 0$. We need to find the function $f(

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

1. The problem is to find the Wendepunkt (inflection point) of the function $f(x) = 3x e^{-0.5x + 1}$.\n\n2. To find the inflection point, we need to find where the second derivati

1. The problem is to find the Wendepunkt (inflection point) of a function, which is where the curvature changes sign. 2. To find the Wendepunkt, we need to find where the second de

1. The problem is to evaluate the definite integral $$\int_{-3}^{3} \sqrt{9 - x^2} \, dx$$. 2. This integral represents the area under the curve of the function $$y = \sqrt{9 - x^2

1. The problem is to evaluate the definite integral $$\int_{-3}^{0.3} \sqrt{9 - x^2} \, dx.$$\n\n2. This integral represents the area under the curve of the function $y = \sqrt{9 -

1. The problem is to evaluate the definite integral $$\int_{-3}^2 (3\sqrt{9-x^2}) \, dx.$$\n\n2. This integral represents the area under the curve of the function $f(x) = 3\sqrt{9-

1. **State the problem:** Find the limit as $n$ approaches infinity of the expression $$\lim_{n \to \infty} \frac{\sqrt{(4n+1)(5+n)(4+n)}}{3n^2 + 2n + 11}.$$\n\n2. **Rewrite the ex

1. مسئله الف: محاسبه مشتق تابع $$y = \ln\left(\sin\left(\sqrt{3x - 1}\right)\right)$$. 2. فرمول‌های مورد استفاده:

1. נניח שהבעיה היא למצוא את הנקודות שבהן הנגזרת של הפונקציה $f(x) = \ln(x) - x$ שווה ל-0. 2. נחשב את הנגזרת של הפונקציה: