∫ calculus

1. **Problem statement:** Determine whether the improper integral $$\int_1^{\infty} \frac{\ln x}{x} \, dx$$ converges or diverges. 2. **Recall the integral test and behavior of fun

1. **State the problem:** Calculate the integral $$\int x \sin x \, dx$$. 2. **Formula and method:** Use integration by parts, which states:

1. The problem is to find the limit: $$\lim_{n \to \infty} \frac{1}{\ln n}$$. 2. Recall that the natural logarithm function $\ln n$ increases without bound as $n$ approaches infini

1. The problem is to find the derivative of the function $$f(x) = x \sqrt{4 - x}$$ and explain how to add fractions together into one fraction during the process. 2. First, recall

1. **State the problem:** Find the derivative of the function $$f(x) = x \sqrt{4 - x}$$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rul

1. **Problem Statement:** Find the relative maxima and minima of the function $f$ based on the given graph description.

1. State the problem: Compute the indefinite integral $\int x e^{x^{2}-3}\,dx$.\n2. Method and formula: We use $u$-substitution because the integrand is $x$ times a function of $x^

1. **State the problem:** We want to evaluate the integral $$\int \frac{x \, dx}{\sqrt{49 + x^4}}.$$\n\n2. **Identify the integral type and substitution:** Notice the expression un

1. **State the problem:** Evaluate the integral $$\int \frac{20 \, dx}{(100x^2 + 1)^2}$$. 2. **Rewrite the integral:** Let us set $$a^2 = \frac{1}{100}$$ so that $$a = \frac{1}{10}

1. **State the problem:** Evaluate the definite integral $$\int_0^{3\sqrt{2}} \frac{dx}{(36 - x^2)^{3/2}}.$$\n\n2. **Recall the formula:** The integral $$\int \frac{dx}{(a^2 - x^2)

1. **State the problem:** Evaluate the definite integral $$\int_0^{3\sqrt{2}} \frac{dx}{(36 - x^2)^{3/2}}.$$\n\n2. **Recall the formula:** The integral $$\int \frac{dx}{(a^2 - x^2)

1. **State the problem:** Evaluate the integral $$\int \frac{x}{\sqrt{81x^2 - 1}} \, dx$$. 2. **Identify the integral type:** This is an integral involving a rational function with

1. **State the problem:** We want to evaluate the integral $$\int \tan^5 x \sec^2 x \, dx.$$\n\n2. **Recall the formula and substitution:** Notice that the derivative of $\tan x$ i

1. **State the problem:** Evaluate the integral $$\int \frac{\sqrt{y^2 - 81}}{y} \, dy$$ for $$y > 9$$. 2. **Identify substitution:** Since the integrand contains $$\sqrt{y^2 - 81}

1. The problem is to find the differential $dy$ when using the substitution $y = 9 \sec \theta$. 2. Recall the substitution: $$y = 9 \sec \theta$$

1. The problem is to find which substitution transforms the integral $$\int \frac{\sqrt{y^2 - 81}}{y} \, dy$$ with the condition $y > 9$ into an integral that can be evaluated dire

1. **State the problem:** Evaluate the integral $$\int \frac{dx}{\sqrt{49x^2 - 81}}$$ with the condition $$x > \frac{9}{7}$$. 2. **Identify the form:** The integral is of the form

1. We are asked to evaluate the integral $$\int \sqrt{4 - t^2} \, dt$$. 2. This integral represents the area under the curve of the function $$y = \sqrt{4 - t^2}$$, which is the up

1. **State the problem:** Evaluate the integral $$\int_0^{\frac{5}{\sqrt{2}}} \frac{dx}{\sqrt{25 - x^2}}.$$\n\n2. **Identify the substitution:** The integral has the form $$\int \f

1. **State the problem:** We need to evaluate the integral $$\int_0^{\frac{5}{\sqrt{2}}} \frac{dx}{\sqrt{25 - x^2}}$$ and determine which substitution transforms it into an integra

1. **State the problem:** Evaluate the integral $$\int_0^2 \frac{dx}{\sqrt{16 - x^2}}.$$\n\n2. **Choose substitution:** We recognize the integrand involves $$\sqrt{a^2 - x^2}$$ wit