নির্দিষ্ট যোগজ

Let I=∫π/4π/3sin⁡xxdxI=\displaystyle \int _{ \pi /4 }^{ \pi /3 }{ \cfrac { \sin { x } }{ x } } dx. Then?

হানি নাটস

I=∫π4π3sinxxdxI=\displaystyle \int^{\tfrac{\pi}{3}}_{\tfrac{\pi}{4}}\dfrac{sinx}{x}dx

sinxx\dfrac{sinx}{x} is a decreasing function in given interval

difference of limits =π3−π4=π12=\dfrac{\pi}{3}-\dfrac{\pi}{4}=\dfrac{\pi}{12}

so, π12⋅sinπ3π3≤I≤π12⋅sinπ4π4\dfrac{\pi}{12}\cdot\dfrac{sin\dfrac{\pi}{3}}{\dfrac{\pi}{3}} \le I \le \dfrac{\pi}{12} \cdot\dfrac{sin\dfrac{\pi}{4}}{\dfrac{\pi}{4}}

38≤I≤26\dfrac{\sqrt3}{8} \leq I \leq \dfrac{\sqrt2}{6}

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Related question

The value of ∫−π/2199π/2(1+cos⁡2x)dx\displaystyle\int^{199\pi/2}_{-\pi/2}\sqrt{(1+\cos 2x)}dx is?

∫0π6sin⁡2xcos⁡xdx= \int_{0}^{\frac{\pi}{6}} \sin ^{2} x \cos x d x= কত ?

 f(x)=  {x+1forx=0 \left \lbrace \begin{matrix} x + 1 & f{\quad\text{or}\quad} & x & = & 0 \end{matrix} \right .  হলে-

  1.   ∫−1−12f(x)dx=18 \int_{- 1}^{- \frac{1}{2}} f{\left ( x \right )} dx = \frac{1}{8}

  2.   ∫01f(x)dx=0 \int_{0}^{1} f{\left ( x \right )} dx = 0

  3.   f(−1)=1 f{\left ( - 1 \right )} = 1

নিচের কোনটি সঠিক?

The value of ;∫0π/4sec⁡x(sec⁡x+tan⁡x)2dx\displaystyle \int_{0}^{\pi /4}\frac{\sec x}{\left ( \sec x+\tan x \right )^{2}}dx is& ;