∫ calculus

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

1. **State the problem:** Evaluate the integral $$\int \frac{dx}{\sqrt{529 + x^2}}$$. 2. **Recall the formula:** The integral of the form $$\int \frac{dx}{\sqrt{a^2 + x^2}}$$ is $$

1. **State the problem:** Evaluate the integral $$\int 13 \cot^4 x \, dx$$. 2. **Recall the formula and identities:** We know that $$\cot^2 x = \csc^2 x - 1$$ and the integral of p

1. **State the problem:** Evaluate the integral $$\int 12 \tan^3(x) \, dx$$. 2. **Recall the formula and rules:** We use the identity $$\tan^3(x) = \tan(x) \cdot \tan^2(x)$$ and th

1. **State the problem:** We want to evaluate the integral $$\int 9 \sec^4 x \tan^8 x \, dx.$$\n\n2. **Recall formulas and identities:** We use the identity $$\sec^2 x = 1 + \tan^2

1. **State the problem:** Evaluate the integral $$\int \sec^2(6x) \tan^5(6x) \, dx$$. 2. **Recall the formula and substitution:** We know that $$\frac{d}{dx} \tan(6x) = 6 \sec^2(6x