∫ calculus

1. **State the problem:** We have a curve passing through the point $(1,-5)$ with a gradient function (derivative) given by $\frac{dy}{dx} = 4x^3$. We need to find the value of $x$

1. **State the problem:** We need to evaluate the definite integral $$\int_1^4 x^2 \ln x \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **Problem statement:** Find the derivative of the function $$y = \log_5(7x) + 8^{9x}$$. 2. **Recall the formulas and rules:**

1. **State the problem:** Calculate the definite integral $$\int_1^8 x \, dx$$ which represents the area under the curve of the function $y = x$ from $x=1$ to $x=8$. 2. **Formula a

1. The problem is to understand the accuracy of calculus solutions provided. 2. Calculus involves limits, derivatives, integrals, and series expansions, which require precise appli

1. **State the problem:** Compute the indefinite integral $$\int \frac{1}{\sqrt{t}} + \sqrt[3]{t} \, dt$$ using the substitution $$t = u^6$$ and then applying partial fractions dec

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $f(x) = \sin(x) + \cos(x)$.\n\n2. **Recall the derivative rules:** The derivative of $\sin(x)$ is $\co

1. The problem is to find the derivative $\frac{dy}{dx}$ of the function $f(x) = \ln(x^4)$.\n\n2. Recall the chain rule and the derivative of the natural logarithm function: if $f(

1. **State the problem:** Find the derivative with respect to $x$ of the function $y = \frac{\tan(x)}{x}$. 2. **Recall the formula:** To differentiate a quotient $\frac{u}{v}$, use

1. **Problem Statement:** We need to find the sign diagram for the derivative of the function $$y = x + \frac{1}{x}$$. 2. **Find the derivative:** Use the power rule and the deriva

1. **Problem Statement:** We want to minimize the time function $T$ by finding the critical points where its derivative $\frac{dT}{dx}$ equals zero. 2. **Given Derivative Expressio

1. **State the problem:** We are given the derivative of a function $g(x)$ as $$\frac{dg}{dx} = \frac{3}{\sqrt{x}} \cdot \left(1 - 2x^2\right)$$ and the initial condition $g(8) = 1

1. **State the problem:** Find the average value of the function $g(x) = -x \cdot e^{x^2}$ on the interval $[1, 3]$. 2. **Formula for average value:** The average value $A$ of a co

1. **State the problem:** Calculate the definite integral $$\int_0^\pi \sin(t) \cdot \cos(t) \, dt.$$\n\n2. **Recall the formula and rules:** We can use the substitution or a trigo

1. **State the problem:** Evaluate the definite integral $$\int_0^{\frac{\pi}{4}} \tan(x) \, dx.$$\n\n2. **Recall the formula:** The integral of $$\tan(x)$$ is $$-\ln|\cos(x)| + C$

1. **State the problem:** Evaluate the definite integral $$\int_0^1 x \sqrt{1 - x^2} \, dx$$ using a change of variables. 2. **Choose substitution:** Let $$u = 1 - x^2$$. Then, dif

1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{x}{\sqrt{1+x^2}} \, dx.$$\n\n2. **Use substitution:** Let $$u = 1 + x^2.$$ Then, differentiate both sides

1. **State the problem:** Evaluate the integral $$\int \frac{x^2}{(x^3 - 3)^2} \, dx.$$\n\n2. **Identify substitution:** Notice the denominator is $(x^3 - 3)^2$ and the numerator i

1. **Stating the problem:** Evaluate the definite integral $$\int_{-2}^2 \left(x^3 \cos(x^2) + \frac{1}{2}\right)^{4 - x^2} \, dx = \pi.$$\n\n2. **Understanding the integral:** The

1. **State the problem:** We need to evaluate the definite integral $$\int_{-2}^{2} \left(x^3 \cos\left(\frac{x}{2}\right) + \frac{1}{2}\right) \sqrt{4 - x^2} \, dx$$.

1. **State the problem:** We want to find the indefinite integral $$\int (11x - 7)^{-3} \, dx$$ using a suitable change of variables. 2. **Formula and substitution:** When integrat