∫ calculus

1. **State the problem:** We want to evaluate the definite integral $$\int_0^a \sqrt{b^2 - \frac{x^2 b^2}{a^2}} \, dx.$$ 2. **Simplify the integrand:** Factor out $b^2$ inside the

1. **Stating the problem:** You want to integrate a function that is squared, but you cannot or do not want to take the square root. 2. **General approach:** When integrating a squ

1. The problem asks for the second derivative of the function $$f(x) = x^7 + x^3 - 21$$ evaluated at $$x=1$$. 2. Recall the rules for derivatives:

1. **Problem:** Given the graph of a function $f$, find the value of $k$ where $f$ is defined at $k$ but $\lim_{x \to k} f(x)$ does not exist. 2. **Understanding the problem:** The

1. **State the problem:** Evaluate the integral $$\int (3x^2 - 4x + 5) \, dx$$. 2. **Recall the formula:** The integral of a power function $$x^n$$ is given by $$\int x^n \, dx = \

1. The problem is to find the integral of $\sin 2x$ with respect to $x$, i.e., $\int \sin 2x \, dx$. 2. Recall the formula for integrating sine functions: $\int \sin(ax) \, dx = -\

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 have a triangle with two vertices fixed on the x-axis at points $(1,0)$ and $(5,0)$. The third vertex lies on the curve defined by $$y = \ln(2x) - \fra

1. **State the problem:** We want to find which of the given limits equals the definite integral $$\int_2^5 x^2 \, dx$$. 2. **Recall the definition of a definite integral as a limi

1. **State the problem:** Find the equation of the tangent line to the graph of $f(x)=\sqrt{2x^2+1}$ at $x=-1$. 2. **Recall the formula for the tangent line:** The equation of the

1. **State the problem:** We are given a velocity function $v(t)$ over the interval $0 \leq t \leq 5$ with known displacement and total distance traveled. We need to find the value

1. **Problem statement:** Find the number of points of inflection of the function $f$ on the interval $0 < x < 3$ given that $f''(x) = \sin(3x) - \cos(x^2)$. 2. **Recall:** Points

1. The problem asks to find the locations of discontinuities for four different graphs described. 2. Discontinuities in functions often occur where the function is undefined, such

1. The problem involves identifying the locations of removable discontinuities (holes) and vertical asymptotes in given graphs. 2. A removable discontinuity (hole) occurs where a f

1. **State the problem:** Find the derivative of the function $$y = \ln(\cos(x))^x$$ with respect to $$x$$. 2. **Rewrite the function:** The function can be written as $$y = (\ln(\

1. Énonçons le problème : Calculer le volume total donné par $$V = \int_{\pi/2}^{3\pi/2} -2\pi x \cos(x) \, dx$$

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form li

1. The problem is to find the domain of the function $f(x,y) = \ln(xy)$.\n\n2. The natural logarithm function $\ln(z)$ is defined only for $z > 0$. This means the argument inside t

1. **State the problem:** We want to find the volume $V$ by integrating the volume element $dV = \pi r^2 dh$ where $r = y = \frac{2}{x}$ and $dh = dx$. 2. **Write the formula:** Th

1. The problem is to differentiate the function $$f(x) = \frac{3x^2 - x}{\sqrt{1-2x}}$$. 2. We use the quotient rule for derivatives: $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}

1. **Problem 1: Find the limit of the sequence** $$a_n = \frac{(3n + 1)!}{(3n - 1)!}$$ The factorial expression can be expanded: