Calculus Practice Set 5

Let $f(x)=x^2e^x$, find $f^{(10)}(0)$.

Calculate: $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\frac{1+\sin x}{1+\cos x}\,dx$

Given $\begin{cases} x=\cos 2t \\ y=(1-\cos t)^2 \end{cases}$, find $\frac{d^2y}{dx^2}$.

Calculate: $f(x)=\int_0^x e^{-\frac{t^2}{2}}\,dt$, $x\in(-\infty,+\infty)$, find asymptotes of curve $y=f(x)$ (if they exist).

Find: $\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$

Given $f(x)$ is continuous and $f(x+2)-f(x)=x$, $\int_0^2 f(x)\,dx=1$, find $\int_1^3 f(x)\,dx$.

Find: $\lim_{x\to 0}\frac{(\cos x)^2-1}{x}$

Calculate indefinite integral: $\int\frac{x^3}{\sqrt{x^2+1}}\,dx$

Calculate improper integral: $\int_0^{+\infty}\frac{dx}{1+x^4}$

Given $f(x)=xe^x-y+\frac{1}{2}\sin y=0$, find $y|_{x=0}$ and $y'|_{x=0}$.

Find curve $y=e^x$, $0\leq x\leq\ln\sqrt{3}$ arc length.