∫ calculus

1. The problem is to find the integral of $\tan^6(x)$ with respect to $x$. 2. We use the identity $\tan^2(x) = \sec^2(x) - 1$ to rewrite powers of tangent in terms of secant.

1. **Problem statement:** We have the line $y = 3x + 4$ and shaded rectangles under the line from $x=0$ to $x=50$. Each rectangle's height is the function value at the left side of

1. We are asked to find the integrals: $$I_1 = \int 2xe^x \, dx, \quad I_2 = \int 2x \cos x \, dx, \quad I_3 = \int 2x \sin x \, dx, \quad I_4 = \int 2x \ln x \, dx$$

1. **Problem statement:** Find the integrals \(I_1 = \int 2x e^x \, dx\) and \(I_4 = \int 2x \ln x \, dx\).\n\n2. **Formula and rules:** For integrals involving products, use integ

1. **Problem:** Find the integral $$I_1 = \int 2x e^x \, dx$$. 2. **Formula and method:** Use integration by parts, which states:

1. **Problem:** Find the limit $$\lim_{x \to -\infty} \frac{\sqrt{x^2+5}}{x-\#}$$ where \# is a constant. 2. **Formula and rules:** For limits involving infinity and square roots o

1. **Problem Statement:** List all intervals on which the given function is continuous, knowing it is discontinuous only at $x=0$ and $x=28$.

1. **State the problem:** Find the limit $$\lim_{n \to \infty} \left(\frac{2n + 10}{3n + 2}\right)^n.$$\n\n2. **Rewrite the expression inside the limit:** Simplify the fraction ins

1. **State the problem:** Calculate the definite integral $$\int_0^1 x^{\frac{1}{2}} (1-x)^{\frac{1}{3}} \, dx$$.

1. **State the problem:** Given a twice-differentiable function $f(x)$ with $f''(x) = 0$ for all $x$ in the domain, determine which statements must be true. 2. **Recall the meaning

1. We need to calculate the definite integral $$\int_{-2}^{2} (4 - x^2)(4 + x^2) \, dx$$. 2. First, expand the integrand using the distributive property:

1. We are asked to calculate the definite integral of the function $f(x) = (4 - x^2)(4 + x^2)$ from $x = -2$ to $x = 2$. 2. First, expand the product inside the integral:

1. **State the problem:** Find the limit $$\lim_{x\to0} \frac{\sin 5x}{3x}$$. 2. **Recall the standard limit formula:** $$\lim_{x\to0} \frac{\sin x}{x} = 1$$.

1. **State the problem:** We need to evaluate the definite integral $$\int_0^1 \left(0.16(\log(x))^2 + 0.704\log(x) + 0.7744\right) dx.$$ 2. **Recall useful formulas:** For $a > -1

1. **Problem:** Determine the intervals where the function $f(x) = x^3 - 3x^2 + 4$ is concave up or concave down. 2. **Formula and rules:**

1. **State the problem:** We are given the integral

1. The problem asks for the equation of a function with two stationary points and a point of inflection. 2. A stationary point occurs where the first derivative of the function equ

1. Problem: Find the derivatives of the given functions. 2. Recall the power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ and the derivative of $$x^x$$ using logarithmic differentiation.

1. Problem: Find the derivatives of the given functions. 2. Recall the power rule for derivatives: $$\frac{d}{dx} x^n = n x^{n-1}$$ and the derivative of a constant is 0.

1. Problem: Differentiate the function $f(x) = x^6 + 2x - 1$. 2. Use the power rule for derivatives: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

1. **State the problem:** We want to evaluate the integral $$\int \frac{x + 4}{(x - 9)(x^2 + 4)} \, dx$$