Found problems: 713
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx$.
For $ f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}$, evaluate $ \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
Evaluate the integrals as follows.
(1) $\int \frac{x^2}{2-x}\ dx$
(2) $\int \sqrt[3]{x^5+x^3}\ dx$
(3) $\int_0^1 (1-x)\cos \pi x\ dx$
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate
$\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$
Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$.
Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.
Evaluate
\[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]
2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$.
Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$.
[i]2011 Tokyo University of Science entrance exam/Management[/i]
Find the area enclosed by the graph of $a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).$
Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality.
\[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]
Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.
Calculate the following indefinite integrals.
[1] $\int \sin (\ln x)dx$
[2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$
[3] $\int \frac{x^3}{x^2+1}dx$
[4] $\int \frac{x^2}{\sqrt{2x-1}}dx$
[5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$.
(1) Find $ I_1,\ I_2$.
(2) Find $ \lim_{n\to\infty} I_n$.
Find the limit as below.
\[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]
Find the following limit.
$ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.
Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$
[i]2010 Hirosaki University School of Medicine entrance exam[/i]
Prove that for each positive integer $n$
\[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$.
Answer the following questions.
(1) Find $ a_1$.
(2) Express $ a_{n\plus{}1}$ in terms of $ a_n$.
(3) Find $ \lim_{n\to\infty} a_n$.
(4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.
Evaluate $\int_0^{\infty} \frac{1}{e^{x}(1+e^{4x})}dx.$
Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.