How do you Integrate ?

Question

$int_0^(2x)sqrt(1+sin(x/2))dx$

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Answer 1 · 2018-05-04T19:04:59

$sin(x/2)+cos(x/2)-1$

$int_0^(2x)sqrt(1+sin(x/2))*dx$

$color(green)(sin(x/2)=2sin(x/4)cos(x/4)$

$color(green)(1=sin^2(x/4)+cos^2(x/4)$

Substitute

$int_0^(2x)sqrt(1+sin(x/2))*dx$

$=int_0^(2x)sqrt(sin^2(x/4)+2sin(x/4)cos(x/4)+cos^2(x/4))*dx$

By Completing The Square

$=int_0^(2x)sqrt((sin(x/4)+cos(x/4))^2)*dx$

$=int_0^(2x)(sin(x/4)+cos(x/4))*dx$

$=[sin(x/4)+cos(x/4)]_0^(2x)$

$=sin(x/2)+cos(x/2)-1$

$color(blue)("if this question was like this " sqrt(1+cos(x/2)) " it can be simplified to look like this"$

$color(blue)(sqrt2*cos(x/4) "using the Double angles Formulae"$