Find the exact length of the curve $y=\frac{1}{4}{x}^2-\frac{1}{2}\ln \left(x\right)$ for $1\le x\le 4$.

Evaluate the integral $\int \frac{5x^2\sqrt{x}-4}{x^2} dx$.

Which of the following is a power series representation for the function $f\left(x\right)=x^6\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following best describes the contour lines (level curves) of the function $f(x,y)=x^2+7y^2$?

Let $r\left(t\right)$ be the vector-valued function defined by $r\left(t\right)=(3t,\frac{1}{t+1},e^{-t})$. Evaluate the definite integral from $t=0$ to $t=1$ of $r\left(t\right)dt$.

Evaluate the line integral of the vector field $F(x,y)=(2x,3y)$ along the curve $C$ given by the vector function $r\left(t\right)=(t,t^2)$, where $t$ goes from $0$ to $1$.

Find the exact length of the polar curve $r=2$, for $0\le \theta \le 4\pi$.

How does the graph of the catenary y = a cosh x/a change as a > 0 increases?

101–104. Proving identities Prove the following identities.

cosh (x + y) = cosh x cosh y + sinh x sinh y