∫ calculus

1. **State the problem:** We want to approximate the area under the curve $f(x) = x^2$ on the interval $[0,1]$ using the left Riemann sum with $n=4$ subintervals. 2. **Formula and

1. **Problem Statement:** Compute the Riemann sum $$\sum_{i=1}^n \left(1 + \frac{2i}{n}\right) \frac{1}{n}$$ and find its limit as $$n \to \infty$$. 2. **Understanding the Riemann

1. **State the problem:** We need to evaluate the integral $$\int x^2 \sqrt{x-1} \, dx$$ using the method of substitution. 2. **Choose a substitution:** Let $$u = x - 1$$. Then, $$

1. مسئله: یافتن فرمول انتگرال تابع داده شده است. 2. فرمول کلی انتگرال: اگر تابعی به صورت $f(x)$ باشد، انتگرال آن به صورت $$\int f(x) \, dx = F(x) + C$$ است که در آن $F'(x) = f(x)$

1. **State the problem:** We want to evaluate the integral $$\int x^2 \sqrt{x+1} \, dx$$ using substitution. 2. **Choose substitution:** Let $$u = x + 1$$. Then, $$du = dx$$ and $$

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{x-1} \, dx$$. 2. **Formula and approach:** When integrating a rational function where the degree of the

1. **State the problem:** Evaluate the integral $$\int \tan^4 x \sec x \, dx$$. 2. **Recall relevant identities:**

1. **State the problem:** We need to solve the integral $$\int x^4 (3 - 5x^5)^{\frac{1}{3}} \, dx$$ using the method of substitution. 2. **Identify substitution:** Let $$u = 3 - 5x

1. **State the problem:** Evaluate the integral $$\int \tan^4 x \sec x \, dx$$. 2. **Recall relevant identities and formulas:**

1. **State the problem:** Find the differential coefficient (derivative) of the function $y = e^{\sin x}$. 2. **Recall the formula:** The derivative of an exponential function $e^{

1. **Problem Statement:** Consider the function $$P(x) = \frac{x^2}{x^2 + 4}$$.

1. **Problem Statement:** Determine all intervals on which the graph of the function $f$ is increasing. 2. **Understanding Increasing Intervals:** A function is increasing on inter

1. The problem asks for the partial derivative of the function $$f(x, y) = 3x^2 + 4xy + y^2$$ with respect to $$y$$. 2. The formula for the partial derivative of a function $$f(x,

1. **State the problem:** We are given the function $$f(x,y) = \sin(xy) + x^2 \ln(y)$$ and need to find the mixed partial derivative $$f_{yx}(0, \frac{\pi}{2})$$, which means first

1. **State the problem:** We are given the function $z = 3xy + 4x^2$ and asked to find the partial derivative of $z$ with respect to $y$. 2. **Recall the formula:** The partial der

1. The problem is to find the derivative of the function $$f(x) = e^{x^2}$$. 2. We use the chain rule for differentiation, which states that if $$f(x) = e^{g(x)}$$, then $$f'(x) =

1. **Problem Statement:** Find the derivative of the function $f(x) = 4 \log x^3$. 2. **Rewrite the function:** Using logarithm properties, $\log x^3 = 3 \log x$, so

1. **State the problem:** Find the second derivative of the function $$f(x) = \frac{2x^2 + 3x + 4}{x}$$. 2. **Rewrite the function:** Simplify the expression by dividing each term

1. **Problem Statement:** We are given the function $$F(x) = x^3 - 6x^2 + 9x + 1$$ and need to analyze its behavior by finding derivatives, critical points, inflection points, inte

1. The problem is to evaluate the definite integral $$\int_3^5 \frac{dx}{x^5 + 1}$$. 2. The integral involves a rational function with a polynomial in the denominator. There is no

1. **Problem Statement:** Given the graph of the derivative $f'(x)$ of a continuous function $f$, determine intervals where $f$ is increasing or decreasing, find local maxima and m