∫ calculus

1. Problem: Evaluate the integral $$- \int_0^1 x^2 (1 - x)^3 \, dx$$. 2. Formula: Use the binomial expansion for $(1-x)^3 = 1 - 3x + 3x^2 - x^3$.

1. **Problem 1:** Evaluate the definite integral $$\int_1^0 (3x+2)^4 \, dx$$. - We use the substitution method. Let $$u = 3x + 2$$, then $$du = 3 \, dx$$ or $$dx = \frac{du}{3}$$.

1. **State the problem:** Find the definite integral $$\int_0^2 (2x - 1)^3 \, dx$$ using integration by substitution.

1. The problem is to complete the blanks in the explanation of Integration by Substitution for Definite Integrals. 2. From the given formula:

1. The problem is to find the indefinite integral of the function $$\frac{4}{4x} + x^3$$ with respect to $x$. 2. We can rewrite the function inside the integral as $$\frac{4}{4x} +

1. Problem: Find $\frac{dy}{dx}$ for the exponential function $y = e^{3x^2}$. 2. Formula: The derivative of $e^{u(x)}$ with respect to $x$ is given by the chain rule:

1. مسئله را بیان می‌کنیم: باید حد تابع $$f(x) = x \left\lceil \frac{1}{x} \right\rceil$$ را وقتی $$x \to 0$$ محاسبه کنیم. 2. تابع شامل قسمت صحیح بالا (سقف) است که برای هر عدد $$y$$

1. **Problem statement:** Evaluate the definite integral $$\int_1^2 (5x^6 + 2x - 1) \, dx$$ and the indefinite integral $$\int 5x^3 \, dx$$. 2. **Formula and rules:**

1. **Stating the problem:** We are given the differential equation $$y' = k + e^{\frac{a}{r}}$$ and asked to find the function $$y$$ by integrating with respect to $$w$$.

1. The problem is to find the integral of the expression $(2x+4)\sqrt{2x^2+3x+1}$. 2. We use the substitution method for integration. Let $u = 2x^2 + 3x + 1$. Then, calculate $\fra

1. **State the problem:** Find the stationary points of the function $$y = x^3 + \frac{11}{2}x^2 - 4x + 31$$ and classify them using a sign diagram and the second derivative test.

1. Problem: Find the derivative $f'(x)$ if $f(x) = 8x^3 + 9x^2 - 5$. 2. Formula: The derivative of $x^n$ is $nx^{n-1}$.

1. **Problem Statement:** Find the derivative of a function $f(x)$ using the basic rules of calculus. 2. **Formula Used:** The derivative of a function $f(x)$ is given by the limit

1. **Problem Statement:** Determine the one-sided and two-sided limits of the function $f(x)$ at specified points using the graph description.

1. Problem A: Integrate $\int x e^{2x} \, dx$ using integration by parts with $u = x$ and $dv = e^{2x} \, dx$. 2. Recall the integration by parts formula:

1. **State the problem:** We want to evaluate the integral $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$$. 2. **Use substitution:** Let $$u = \sqrt{x} = x^{1/2}$$. Then, $$x = u^2$$ an

1. **State the problem:** We need to find the integral $$\int (x^2 + 3x)^2 \, dx$$. 2. **Use the formula:** To integrate a polynomial squared, first expand the expression inside th

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^4 - 3}{x^2} \, dx$$. 2. **Rewrite the integrand:** Simplify the fraction by dividing each term in the nume

1. **State the problem:** Evaluate the integral $$\int \frac{\sin x + \cos x}{\sin x + \cos x} \, dx$$. 2. **Simplify the integrand:** Since the numerator and denominator are the s

1. **State the problem:** We need to evaluate the integral $$\int x(3x^2 + 5)\,dx$$. 2. **Use the distributive property:** Multiply $x$ inside the parentheses:

1. مسئله: مشتق توابع داده شده را با استفاده از تعریف مشتق حساب کنید. 2. تعریف مشتق: مشتق تابع $f(x)$ در نقطه $x$ برابر است با حد