If M -0- - 2cos x - x -2 dx - N -0- - 4sin x cos x - x -12 dx- then the value of M - N isA -D- 4-4-2

The correct option is C 2π + 4We know that 2sin xcos x=sin2x, N = ∫^π/4_0sin2x/2(x+1)^2dx Let 2x = t ⇒ 2dx = dt ⇒ N = ∫^π/2_0 sin t/4 (t/2 + 1 )^2 dt ⇒ N = ∫^π ...

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Byju's Answer

Standard XII

Mathematics

Property 5

If M =∫0π / 2...

Question

If $M = π / 2 \int 0 \frac{cos x}{x + 2} d x, N = π / 4 \int 0 \frac{sin x cos x}{(x + 1)^{2}} d x,$ then the value of $M - N$ is

π

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\frac{π}{4}

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\frac{2}{π - 4}

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\frac{2}{π + 4}

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Solution

The correct option is D $\frac{2}{π + 4}$
We know that $2 sin x cos x = sin 2 x$ ,
$N = π / 4 \int 0 \frac{sin 2 x}{2 (x + 1)^{2}} d x$
Let
$2 x = t \Rightarrow 2 d x = d t$
$\Rightarrow N = π / 2 \int 0 \frac{sin t}{4 {(\frac{t}{2} + 1)}^{2}} d t \Rightarrow N = π / 2 \int 0 \frac{sin t}{(t + 2)^{2}} d t$
Using integration by parts we have
$\Rightarrow N = sin t (- \frac{1}{t + 2}) {∣ ∣ ∣}_{t = 0}^{t = \frac{π}{2}} + π / 2 \int 0 \frac{cos t}{t + 2} d t$
$\Rightarrow N = - \frac{2}{π + 4} + M \Rightarrow M - N = \frac{2}{π + 4}$

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If M=π/2∫0cosxx+2 dx,N=π/4∫0sinxcosx(x+1)2dx, then the value of M–N is

The correct option is D 2π+4We know that 2sinxcosx=sin2x, N=π/4∫0sin2x2(x+1)2dx Let 2x=t⇒2dx=dt ⇒N=π/2∫0sint4(t2+1)2dt⇒N=π/2∫0sint(t+2)2dt Using integration by parts we have ⇒N=sint(−1t+2)∣∣∣t=π2t=0+π/2∫0costt+2dt ⇒N=−2π+4+M⇒M−N=2π+4

If $M = π / 2 \int 0 \frac{cos x}{x + 2} d x, N = π / 4 \int 0 \frac{sin x cos x}{(x + 1)^{2}} d x,$ then the value of $M - N$ is