∫ calculus

1. **State the problem:** Simplify the integral $$\int \frac{x}{x+1} \, dx$$. 2. **Rewrite the integrand:** Notice that $$\frac{x}{x+1} = \frac{x+1-1}{x+1} = \frac{x+1}{x+1} - \fra

1. **State the problem:** We want to find the integral $$\int \ln(x+1) \, dx$$ using integration by parts. 2. **Recall integration by parts formula:** $$\int u \, dv = uv - \int v

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{x^5 + 2x^2}{e^x - x - 1}$$. 2. **Analyze the expression:** As $x \to 0$, both numerator and denominator a

1. **State the problem:** We want to find the limit $$\lim_{x \to 2} \frac{2 - x}{\sqrt{x+5} - \sqrt{5}}.$$\n\n2. **Rationalize the denominator:** Multiply numerator and denominato

1. **State the problem:** We are given the implicit equation $$4x^2 + y^2 = 4$$ and need to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Differentiate both sides wit

1. **State the problem:** We are given the implicit equation $$\sqrt[3]{x} + \sqrt[3]{y} = 8$$ and need to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Rewrite the e

1. **State the problem:** Given the equation $$x^3 y^3 = 44$$, find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Differentiate both sides with respect to $$x$$:**

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$\ln(2xy) = e^{xy}, \quad y \neq 0.$$\n\n2. **Differentiate both sides with

1. **State the problem:** We want to find $\frac{dy}{dx}$ by implicit differentiation from the equation $$\ln(2xy) = e^{xy}$$ where $y \neq 0$. 2. **Rewrite the equation:** $$\ln(2

1. We are asked to find the integral of $\sqrt{1+u^2}$. 2. Recall the integral formula for $\int \sqrt{a^2 + x^2} \, dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln\left|x +

1. The problem is to find the integral of the function $1 + u^2$ with respect to $u$. 2. Recall that the integral of a sum is the sum of the integrals, so we can write:

1. The problem is to evaluate the integral $$\int \sqrt{1+u^2} \, du$$. 2. To solve this, we use a trigonometric substitution. Let $$u = \sinh(t)$$, so that $$du = \cosh(t) \, dt$$

1. **State the problem:** We need to determine whether the series $$\sum_{n=1}^\infty \left(\sqrt[3]{n^2 + 1} - n\right)$$ converges or diverges. 2. **Analyze the general term:** C

1. The problem is to analyze the given series and understand its behavior and graph. 2. The series is:

1. Problem: Calculate the limit $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4}$$ and similar rational functions. Step 1: Identify the highest power of $n$ i

1. Problem: Calculate the limit $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4}$$ and similar rational functions of polynomials. Step 1: Identify the highest

1. **State the problem:** We want to determine the convergence of the infinite series $$\frac{x}{1 \cdot 2} + \frac{x^2}{3 \cdot 4} + \frac{x^3}{5 \cdot 6} + \cdots$$

1. The problem is to find the integral of $\cosh\left(\frac{x}{a}\right)$ with respect to $x$. 2. Recall that the integral of $\cosh(u)$ with respect to $u$ is $\sinh(u) + C$.

1. Problem: Evaluate the definite integral $$\int_{0}^{1} 5 e^{2x-1} dx$$. 2. Step 1: Factor constants and simplify the integrand by writing $5 e^{2x-1}=5 e^{-1} e^{2x}$.

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{5}{e^{2x-1}} \, dx$$. 2. **Rewrite the integrand:** Since $$\frac{5}{e^{2x-1}} = 5 e^{-(2x-1)} = 5 e^{-2x+

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{2x - 6}{\sqrt{x+1} - 2}$$. 2. **Identify the indeterminate form:** Substitute $x=3$ directly: