MCQ
${I_n} = \int_{\,0}^{\,\pi /4} {{{\tan }^n}x\,dx} $, then $\mathop {\lim }\limits_{n - \infty } n\,[{I_n} + {I_{n - 2}}]$ equals
- A$1/2$
- ✓$1$
- C$\infty $
- D$0$
✓
Answer
Correct option: B.
$1$
b
(b) ${I_n} = \int_0^{\pi /4} {{{\tan }^n}x} \,dx = \int_0^{\pi /4} {\,({{\sec }^2}x - 1){{\tan }^{n - 2}}x\,dx} $
(b) ${I_n} = \int_0^{\pi /4} {{{\tan }^n}x} \,dx = \int_0^{\pi /4} {\,({{\sec }^2}x - 1){{\tan }^{n - 2}}x\,dx} $
$ = \int_0^{\pi /4} {{{\sec }^2}x{{\tan }^{n - 2}}} x\,dx - \int_0^{\pi /4} {{{\tan }^{n - 2}}} x\,dx$
$ = \left[ {\frac{{{{\tan }^{n - 1}}x}}{{n - 1}}} \right]_0^{\pi /4} - {I_{n - 2}}$
==> ${I_n} + {I_{n - 2}} = \frac{1}{{n - \,1}}$
Now, $\mathop {\lim }\limits_{n \to \infty } \,n[{I_n} + {I_{n - 2}}]$
$ = \mathop {\lim }\limits_{n \to \infty } \,\frac{n}{{n - 1}}$
$= \mathop {\lim }\limits_{n \to \infty } \,\frac{1}{{1 - \frac{1}{n}}} = 1$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The interval for which the given function $f(x) = 2{x^3} - 3{x^2} - 36x + 7$ is decreasing, is
→$\left|\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|=?$
→The determinant $\left| {\,\begin{array}{*{20}{c}}{4 + {x^2}}&{ - 6}&{ - 2}\\{ - 6}&{9 + {x^2}}&3\\{ - 2}&3&{1 + {x^2}}\end{array}\,} \right|$ is not divisible by
→Solution of differential equation $\frac{{dy}}{{dx}} = \frac{{y - x}}{{y + x}}$ is
→The function $f : R \rightarrow R$ given by $f(x) = x^3 – 1$ is:
→If $x = a(t + \sin t)$ and $y = a(1 - \cos t)$, then ${{dy} \over {dx}}$ equals
→If the constraints in a linear programming problem are changed:
Solution of differential equation $\frac{{dy}}{{dx}} + ay = {e^{mx}}$ is
→A line $\overrightarrow{O P}$ in space, represented by the figure below; has a magnitude of $2 \sqrt{2}$ units.
Which of these are the direction ratios of $\overrightarrow{O P}$ ?
→Which of these are the direction ratios of $\overrightarrow{O P}$ ?
Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^2=3 A +\alpha I$. If $A ^4=21 A +\beta I$, then
→