Let \(C_n=\displaystyle\int_{\frac{1}{n+1}}^{\frac{1}{n}}\frac{\tan^{-1}(nx)}{\sin^{-1}(nx)}\,dx\), then \(\displaystyle\lim_{n\to\infty}n^2\cdot C_n\) equals
$\frac{1}{2}$
Substitute \(y=nx\), so \(dx=dy/n\); the limits become \(y=\dfrac{n}{n+1}\) to \(y=1\).
So \(C_n=\dfrac1n\displaystyle\int_{n/(n+1)}^{1}\dfrac{\tan^{-1}y}{\sin^{-1}y}\,dy\). Let \(f(y)=\dfrac{\tan^{-1}y}{\sin^{-1}y}\), so \(f(1)=\dfrac{\pi/4}{\pi/2}=\dfrac12\).
By the Mean Value Theorem for integrals, \(\displaystyle\int_{n/(n+1)}^1 f(y)\,dy=f(\xi_n)\left(1-\dfrac{n}{n+1}\right)=f(\xi_n)\cdot\dfrac{1}{n+1}\), for some \(\xi_n\) between \(n/(n+1)\) and 1; as \(n\to\infty\), \(\xi_n\to 1\), so \(f(\xi_n)\to \dfrac12\) by continuity of \(f\) at 1.
Hence \(C_n\approx \dfrac{1}{n}\cdot\dfrac{1}{n+1}\cdot f(\xi_n)\), so \(n^2C_n=\dfrac{n}{n+1}f(\xi_n)\).
Taking \(n\to\infty\): \(\dfrac{n}{n+1}\to 1\) and \(f(\xi_n)\to \dfrac12\), so \(\displaystyle\lim_{n\to\infty}n^2C_n=\dfrac12\).
Similar Questions
If a, b and c are real numbers, then the value of
\(\displaystyle\lim_{t\to 0}\ln\left(\frac{1}{t}\int_{0}^{t}(1+a\sin(bx))^{c/x}\,dx\right)\) equals
The value of the integral
\(\displaystyle\int_{0}^{\infty}\frac{dx}{(1+x^a)(1+x^2)}\ (a>0)\) is
For x > 0, if \(f(x)=\displaystyle\int_1^x \frac{\log_e t}{(1+t)}\,dt\), then f(e) + f(1/e) equals
Let \(a_n=\displaystyle\int_0^{\pi/2}(1-\sin t)^n \sin(2t)\,dt\), then \(\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{a_k}{k}\) is equal to
If [.] represents the Greatest Integer Function, then the value of \(\left|\displaystyle\int_0^{\sqrt{\pi/2}}\left([x^2]-\cos(x)\right)dx\right|\) is
Important Questions from Definite Integrals
The Legendre polynomials P n(x), n = 0, 1, 2, ..., satisfying the orthogonailty condition \(\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{P}}_{\rm{n}}}\left( {\rm{x}} \right){{\rm{P}}_{\rm{m}}}} \left( {\rm{x}} \right){\rm{dx}}\,{\rm{ = }}\,\frac{{\rm{2}}}{{{\rm{2n + 1}}}}{{\rm{\delta }}_{{\rm{nm}}}}\) on the interval [-1, +1], may be defined by the Rodrigues formula P n(x) = \(\frac{{\rm{1}}}{{{{\rm{2}}^{\rm{n}}}{\rm{n!}}}}\frac{{{{\rm{d}}^{\rm{n}}}}}{{{\rm{d}}{{\rm{x}}^{\rm{n}}}}}{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} \right)^{\rm{n}}}\) . The value of the definite integral \(\int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{4 + 2x - 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}} \right){{\rm{P}}_{\rm{3}}}\left( {\rm{x}} \right){\rm{dx}}} \) is
A parametric curve is defined \(x = cos\left(\frac{\Pi t}{2}\right) , Y= sin\left(\frac{\Pi t}{2}\right)\) in the range of \(0\leq t\leq 1\) . It is rotated about X-axis by 360°.
‘What is the area of the surface generated?
if \(\displaystyle\int\dfrac{\sin x}{\sin (x-a)}dx=Ax+B\log |sin(x-a)|+ C\) where A, B and c are real constants then:
Which of the following is NOT a property of definite integral?
The value of integral \(x\over cos^2 x\) is equal to
