-0-2sin 20 x d x-a-b-0-2sin 20 x cos 20 x d x- where a is a prime number- Then the value of a-b is

The correct option is B 22I= ∫_0^π/2sin^20xcos^20x dx = ∫_0^π/2(sin2x2)^20 dx =12^20∫_0^π/2(sin2x)^20 dx Let 2x=t Then I= 1/2^20∫_0^π(sin t)^20·dt2 =12^20∫_ ...

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Standard XII

Mathematics

Integral as Antiderivative

∫0π/2sin 20 x...

Question

$\frac{π}{2} \int 0 {sin}^{20} x d x = a^{- b} \frac{π}{2} \int 0 {sin}^{20} x {cos}^{20} x d x,$ where $a$ is a prime number. Then the value of $a - b$ is

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Solution

The correct option is C $22$
$I = \frac{π}{2} \int 0 {sin}^{20} x {cos}^{20} x d x = \frac{π}{2} \int 0 {(\frac{sin 2 x}{2})}^{20} d x = \frac{1}{2^{20}} \frac{π}{2} \int 0 {(sin 2 x)}^{20} d x$

Let $2 x = t$
Then $I = \frac{1}{2^{20}} π \int 0 {(sin t)}^{20} \cdot \frac{d t}{2}$
$= \frac{1}{2^{20}} \frac{π}{2} \int 0 {sin}^{20} t d t = 2^{- 20} \frac{π}{2} \int 0 {sin}^{20} t d t ∴ a = 2, b = - 20$

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π2∫0sin20x dx=a−bπ2∫0sin20xcos20x dx, where a is a prime number. Then the value of a−b is

The correct option is C 22I=π2∫0sin20xcos20x dx=π2∫0(sin2x2)20 dx=1220π2∫0(sin2x)20 dx Let 2x=t Then I=1220π∫0(sint)20⋅dt2 =1220π2∫0sin20t dt=2−20π2∫0sin20t dt∴a=2,b=−20

$\frac{π}{2} \int 0 {sin}^{20} x d x = a^{- b} \frac{π}{2} \int 0 {sin}^{20} x {cos}^{20} x d x,$ where $a$ is a prime number. Then the value of $a - b$ is