∫ calculus

1. **State the problem:** We need to find the indefinite integral of the function $3x + e^x$ with respect to $x$. 2. **Formula used:** The integral of a sum is the sum of the integ

1. **State the problem:** Evaluate the integral $$\int 6 \cdot 2^{3x} \, dx$$. 2. **Recall the formula:** The integral of an exponential function $$a^{kx}$$ with respect to $$x$$ i

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x}{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivati

1. **State the problem:** We need to find the derivative of the function $$f(x) = x^{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivative:**

1. **State the problem:** We need to find the integral of the function $5e^{2x}$ with respect to $x$. 2. **Recall the formula:** The integral of $e^{ax}$ with respect to $x$ is giv

1. **State the problem:** Evaluate the integral $\int \cos(2x - 4x) \, dx$. 2. **Simplify the argument of cosine:**

1. **State the problem:** We are given the acceleration function of a particle as $a(t) = 36t^2 + 12t$. We need to find the velocity at time $t=2$ given that the velocity at $t=1$

1. The problem states that the derivative of a function $f$ is given by $f'(x) = x^4 + 3x^2 - 2$ and that $f(0) = 4$. We need to find the value of $f(1)$. 2. To find $f(1)$, we fir

1. **Problem statement:** Use the Intermediate Value Theorem (IVT) to verify that $f(x)$ has a zero in the given interval, then use the bisection method to find an interval of leng

1. **State the problem:** Given \( \frac{dy}{dx} = 7x^{2} - 5 \) and \( \frac{dx}{dz} = 6x - 4 \), find \( \frac{dy}{dz} \). 2. **Formula used:** By the chain rule for derivatives,

1. The problem asks to name the discontinuity at $x=2$ for the given function. 2. A discontinuity occurs where a function is not continuous. Common types include removable, jump, a

1. The problem asks to identify the type of discontinuity at $x=2$ based on the graph and given points. 2. A discontinuity occurs where a function is not continuous. Common types a

1. **State the problem:** We are given the curve equation $$y = 2x^3 - x^2 + ax - 5$$ where $$a$$ is a constant. 2. **Find the derivative $$\frac{dy}{dx}$$:** To find the slope of

1. **State the problem:** Differentiate the function $$f(x) = 4x^{\frac{1}{2}} + \frac{3}{2}x^{-\frac{5}{8}}$$ with respect to $$x$$. 2. **Recall the power rule for differentiation

1. **State the problem:** Differentiate the expression $$10r^{5} + 7r^{3} - 11r + 8$$ with respect to $$r$$. 2. **Recall the power rule for differentiation:** For any term $$ar^n$$

1. **State the problem:** We are given a function $g(x)$ and asked to find its derivative $g'(x)$ and then solve for $x$ when $g'(x) = 4$.

1. **Stating the problem:** We need to evaluate the integral $$\int \frac{dx}{x^3 + 8x + 19}$$. 2. **Understanding the integral:** The denominator is a cubic polynomial. To integra

1. **State the problem:** Evaluate the integral $$\int \frac{x^3}{\sqrt{2-3x^4}} \, dx$$. 2. **Identify substitution:** Let $$u = 2 - 3x^4$$. Then, differentiate:

1. **State the problem:** Find the absolute minimum and absolute maximum values of the function $$f(x) = x^3 - 2x^2 - 4x + 1$$ on the closed interval $$[-1, 3]$$. 2. **Formula and

1. **State the problem:** Find the absolute minima and maxima of the function $$f(x) = x^3 - 2x^2 - 4x + 1$$. 2. **Formula and rules:** To find absolute extrema, first find critica

1. **State the problem:** Find the absolute minimum and absolute maximum values of the function $$f(x) = x^3 - 2x^2 - 4x + 1$$ on the closed interval $$[-1, 3]$$. 2. **Formula and