∫ calculus

1. **State the problem:** We are given the integral equation $$\int_0^{x^2} f(t) \, dt = x^2(1 + x)$$ and asked to find the value of $$f(4)$$. 2. **Recall the Fundamental Theorem o

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** This is a limit of a rational function where direct substitu

1. **State the problem:** We want to find the indefinite integral

1. **State the problem:** Evaluate the definite integral $$\int_{-1}^{\alpha} e^{-x} \, dx$$ and write the result to four decimal places. 2. **Recall the formula:** The integral of

1. **State the problem:** Evaluate the definite integral $$\int_1^9 \frac{x - 1}{\sqrt{x}} \, dx$$ which is rewritten as $$\int_1^9 \left(x^{\frac{1}{2}} - x^{-\frac{1}{2}}\right)

1. **State the problem:** Find the limit as $x$ approaches infinity of the function $4x^2 - 7$. 2. **Recall the limit rule for polynomials:** For a polynomial $ax^n + \dots$, as $x

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{9}{4x^2 - 7}$$. 2. **Recall the rule for limits at infinity:** When the degree of the polynomial in the denomin

1. **State the problem:** Find the equation of the tangent line to the function $$f(x) = (1 + 12\sqrt{x})(4 - x^2)$$ at $$x = 9$$ and fill in the blank in the equation $$y = -2849

1. **State the problem:** Find the derivative of $y = 2x \sin x$ and evaluate it at $x = \frac{\pi}{2}$. 2. **Formula and rules:** Use the product rule for derivatives: If $y = u(x

1. **Problem 1:** Find the limit $$\lim_{x \to 3^-} \frac{|x-3|}{x-3}$$. The expression involves the absolute value function and a denominator that approaches zero from the left si

1. **State the problem:** We need to find $\frac{dz}{dt}$ where $z = \sin(x) \cos(y)$, $x = t$, and $y = \frac{1}{t}$. 2. **Formula and rules:** Use the Chain Rule for multivariabl

1. **State the problem:** We are given velocity graphs of two particles and asked to determine intervals when each particle is speeding up or slowing down. 2. **Recall the rule:**

1. State the problem.\nFind $\dfrac{dy}{dx}$ for the parametric equations $y=\dfrac{\theta-1}{\theta+1}$ and $x=\dfrac{\theta^2-1}{\theta^2+1}$.\n\n2. Use the parametric derivative

1. **State the problem:** Find the slope of the tangent line to the function $$f(x) = \frac{4x - 1}{x}$$ at $$x = 1$$. 2. **Recall the formula:** The slope of the tangent line at a

1. **State the problem:** We are given the limit $$\lim_{x \to 1} \frac{h(x) - h(1)}{\ln(x^2)} = 2$$ and asked to find the value of $$h'(1)$$ and analyze if $$h(1)$$ is an extremum

1. **Problem:** Compute $\lim_{x \to c} (f(x) + g(x))$ given $\lim_{x \to c} f(x) = 1$ and $\lim_{x \to c} g(x) = -1$. 2. **Formula:** The limit of a sum is the sum of the limits:

1. **State the problem:** We want to evaluate the limit $$\lim_{x \to 4} (x + 2)$$ using a table of values. 2. **Recall the limit concept:** The limit of a function as $$x$$ approa

1. **State the problem:** We need to find the integral $$\int \frac{1}{x^3 (x^2 + 1)^2} \, dx.$$\n\n2. **Formula and approach:** To integrate rational functions like this, we use p

1. **State the problem:** Find the limit \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \). 2. **Recall the formula and rules:** The expression is a rational function. Direct substitution

1. **Problem:** Find $\frac{dy}{dx}$ using implicit differentiation for the equation: $$x + \sec(y) = \ln(y)$$

1. **State the problem:** We want to find the integral $$I = \int \ln(x^2 + 1) \, dx$$. 2. **Use integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$.