Question #89694

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Answer 1 · 2017-04-14T00:46:31

$int cosxsqrt(1+cos2x)dx =(x +sinxcosx)/sqrt2 + C$

Use the trigonometric identity:

$cos^2x = (1+cos2x)/2$

so:

$cosxsqrt(1+cos2x) = cosx sqrt(2cos^2x) =sqrt2cos^2x = (1+cos2x)/sqrt2$

and:

$int cosxsqrt(1+cos2x)dx = int (1+cos2x)/sqrt2dx$

$int cosxsqrt(1+cos2x)dx = 1/sqrt2 int dx +1/(2sqrt2) int cos2xd(2x)$

$int cosxsqrt(1+cos2x)dx = x/sqrt2 +(sin2x)/(2sqrt2) +C$

$int cosxsqrt(1+cos2x)dx =(x +sinxcosx)/sqrt2 + C$