∫ calculus

1. **State the problem:** We need to evaluate the definite integral $$\int_1^2 x^3 \ln x \, dx$$ and express it in the form $$a c^3 + b$$ where $a$ and $b$ are rational constants.

1. **State the problem:** We want to approximate the definite integral $$\int_{-3}^3 g(x) \, dx$$ using six subintervals and right endpoints. 2. **Determine the width of each subin

1. **State the problem:** We have the curve $C$ defined by $y = x \ln x$ for $x > 0$, and the point $P(e, e)$ on $C$. The line $l$ is the normal to $C$ at $P$. The region $R$ is bo

1. **State the problem:** We have the curve $C$ defined by the equation $$y = x^3 - 10x^2 + 27x - 23$$ and a point $P(5, -13)$ on $C$. We want to find the equation of the tangent l

1. **State the problem:** Gravel is dumped forming a cone-shaped pile with volume increasing at 30 cubic feet per minute. The base diameter equals the height at all times. We need

1. **State the problem:** We want to evaluate the integral $$\int_5^{10} \frac{3 \, dx}{(x - 1)(3 + 2\sqrt{x - 1})}$$ using the substitution $$x = u^2 + 1$$ and determine the new l

1. **State the problem:** We want to evaluate the integral $$\int_5^{10} \frac{3 \, dx}{(x - 1)(3 + 2\sqrt{x - 1})}$$ using the substitution $x = u^2 + 1$.

1. **State the problem:** We are asked to estimate the integral $$\int_{0.5}^{2.5} \sqrt{\frac{x}{1+x}} \, dx$$ using the trapezium rule with given values of $y = \sqrt{\frac{x}{1+

1. **State the problem:** We want to evaluate the integral $$\int_0^6 \frac{x^2 + 8x - 3}{x + 2} \, dx$$ and express the answer in the form $a + b \ln 2$ where $a$ and $b$ are inte

1. **State the problem:** We have the curve $$y=\frac{32}{x^2}+3x-8$$ for $$x>0$$ and point $$P(4,6)$$ lies on this curve. The line $$l$$ is the normal to the curve at $$P$$. The r

1. **State the problem:** Evaluate the limit $$\lim_{x\to 0}\frac{\sin(5x)-5x+\frac{(5x)^3}{6}}{x^5}$$. 2. **Recall the Taylor series expansion for sine:**

1. **State the problem:** Evaluate the limit $$\lim_{x\to 0}\frac{\sin(5x)-5x+\frac{(5x)^3}{6}}{x^5}$$. 2. **Recall the Taylor series expansion for sine:**

1. **State the problem:** We need to evaluate the definite integral $$\int_1^2 x^3 \ln x \, dx$$ and express it in the form $$a c^3 + b$$ where $a$ and $b$ are rational constants.

1. The problem asks if finding the equation of the tangent line means finding the derivative. 2. The equation of the tangent line to a curve at a point involves two things: the slo

1. **State the problem:** We have the function $y = x^2 - k\sqrt{x}$, where $k$ is a constant. We are given the derivative $$\frac{dy}{dx} = 2x - \frac{1}{2}kx^{-\frac{1}{2}}$$

1. **Problem:** Evaluate the integral $$\int \sqrt{x+1} \, dx$$ using substitution. 2. **Step 1: Substitution**

1. The problem is to find the derivative of a function, but the function is not specified. 2. The derivative of a function $f(x)$, denoted $f'(x)$ or $\frac{d}{dx}f(x)$, measures t

1. **State the problem:** We need to find the gradient of the tangent to the curve $y=2x^3+5$ at the point $P(1,7)$ using differentiation from first principles. 2. **Recall the for

1. **State the problem:** We have parametric equations $x(t) = 2 - 4 \sin t$ and $y(t) = 4 \cos t$ for $t$ in $[0, \frac{\pi}{2}]$. We want to find the volume generated when the cu

1. The problem is to find the integral of the function $y=\frac{1}{x^2+1}$.\n\n2. The formula for the integral of $\frac{1}{x^2+1}$ is a standard integral: $$\int \frac{1}{x^2+1} \

1. **Stating the problem:** We want to find the integral $$\int \frac{1}{1+\cos^4 t} \, dt$$ and analyze its behavior for graphing. 2. **Formula and approach:** There is no simple