-0-2sin x sin 2 x sin 3 x sin 4 x d x is equal toA- -4B- -8C- -16D- -32

The correct option is C π/16I=∫_0^π/2sin x sin 2x sin 3x sin 4x dx I=∫_0^π/2 cos x sin2x cos3x sin4xdx 2I=∫_0^π/2sin 2x sin 4x cos 2x =1/2∫_0^π/2sin4x sin 4x ...

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Byju's Answer

Standard XII

Mathematics

Integration Using Substitution

∫0π/2sin x si...

Question

$\int_{0}^{\frac{π}{2}} s i n x s i n 2 x s i n 3 x s i n 4 x d x$ is equal to

\frac{π}{4}

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\frac{π}{8}

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\frac{π}{16}

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\frac{π}{32}

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Solution

The correct option is C $\frac{π}{16}$
$I = \int_{0}^{\frac{π}{2}} s i n x s i n 2 x s i n 3 x s i n 4 x d x$
$I = \int_{0}^{\frac{π}{2}} c o s x s i n 2 x c o s 3 x s i n 4 x d x$
$2 I = \int_{0}^{\frac{π}{2}} s i n 2 x s i n 4 x c o s 2 x$
$= \frac{1}{2} \int_{0}^{\frac{π}{2}} s i n 4 x s i n 4 x$
$2 I = \frac{1}{2} \int_{0}^{\frac{π}{2}} \frac{1 - c o s 8 x}{2} d x = \frac{1}{4} {[x - \frac{s i n 8 x}{8}]}_{0}^{\frac{π}{2}} = \frac{π}{8}$
$\Rightarrow I = \frac{π}{16}$

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∫π20sin xsin2xsin3xsin4x dx is equal to

The correct option is C π16I=∫π20sin xsin2xsin3xsin4xdx I=∫π20cos xsin2xcos3xsin4xdx 2I=∫π20sin2xsin4xcos2x =12∫π20sin4xsin4x 2I=12∫π201−cos8x2dx=14[x−sin8x8]π20=π8 ⇒I=π16

$\int_{0}^{\frac{π}{2}} s i n x s i n 2 x s i n 3 x s i n 4 x d x$ is equal to