Reduction Integration secx (1 Viewer)

gamja · Jan 16, 2023

Evaluate $\int _0^{\frac{\pi }{4}}\:sec^4x$

I know if I(n) = $\int \:cos^n\left(x\right)dx$
then I(n) = $\frac{1}{n}sin\left(x\right)cos^{n-1}\left(x\right)+\frac{n-1}{n}I\left(n-2\right)$

but how would we adapt that to sec^4x?

Thanks for the help in advance!

Drongoski · Jan 16, 2023

Drongoski · Jan 16, 2023

For reduction formula for sec x:

$I_n = \int sec^n x dx = \frac {sec^{n-2}tanx}{n-1} + \frac {n-2}{n-1} \int sec^{n-2} xdx$
$I_4 = \int ^- _- sec^4xdx$

$= \frac {sec^2 x tan x}{4-1} + \frac {4-2}{4-1} \int ^- _- sec^2 x dx \\ \\ = \left [ \frac {sec^2 x tan x}{3} + \frac {2}{3} tan x \right ]^\frac {\pi} {4} _0 \\ \\ = \frac {(\sqrt 2)^2 \times 1}{3} - 0 = \frac {4}{3}$

I don't think you can use the reduction formula for cos^n x.

Reduction Integration secx (1 Viewer)

gamja

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Drongoski

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Drongoski

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