4-0-2cos x-cos x-sin x-

I = 4/π ∫_0^π/2 cos x/cos x + sin xdx — (1) Applying f(x) = f(a x) I = 4/π ∫_0^π/2 cos ( π/2 x )/cos ( π/2 x )+sin ( π/2 r )dx I = 4/π ∫_0^π/2sin x ...

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Standard XII

Physics

Definite Integrals

4/π∫0π/2cos x...

Question

$\frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{c o s x}{c o s x + s i n x} =$ ?

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Solution

$I = \frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{c o s x}{c o s x + s i n x} d x$ ---- (1)
Applying f(x) = f(a - x)
$I = \frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{c o s (\frac{π}{2} - x)}{c o s (\frac{π}{2} - x) + s i n (\frac{π}{2} - r)} d x$
$I = \frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{s i n x}{s i n x + c o s x} d x$ ---- (2)
Adding (1) + (2)
$2 I = \frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{s i n x + c o s x}{s i n x + c o s x} d x 2 I = \frac{4}{π} \int_{0}^{\frac{π}{2}} d x 2 I = \frac{4}{π} \times \frac{π}{2} I = 1$

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4π ∫π20 cos xcos x+sin x= ?

I=4π ∫π20 cos xcos x+sin xdx ---- (1) Applying f(x) = f(a - x) I=4π ∫π20 cos (π2−x)cos (π2−x)+sin(π2−r)dx I=4π ∫π20sin xsin x+cos xdx ---- (2) Adding (1) + (2) 2I=4π ∫π20 sin x+cos xsin x+cos xdx2I=4π∫π20dx2I=4π×π2I=1

$\frac{4}{π} \int_{0}^{\frac{π}{2}} \frac{c o s x}{c o s x + s i n x} =$ ?