cos-x-dx-

Question Number 121466 by sdfg last updated on 08/Nov/20

\int \sqrt{c o s (x) d x}

Commented by MJS_new last updated on 08/Nov/20

\int \sqrt{\cos x d x} or \int \sqrt{\cos x} d x ?

Commented by sdfg last updated on 08/Nov/20

\int \sqrt{\cos x} dx

Answered by MJS_new last updated on 08/Nov/20

\int \sqrt{\cos x} d x =

[t = \frac{x}{2} \to d x = 2 d t]

= 2 \int \sqrt{\cos 2 t} d t = 2 \int \sqrt{{2 \cos}^{2} t - 1} d t =

= 2 \int \sqrt{1 - {2 \sin}^{2} t} d t = 2 E (t ∣ 2) =

= 2 E (\frac{x}{2} ∣ 2) + C

Commented by MJS_new last updated on 08/Nov/20

\int \sqrt{\sin x} d x = \int \sqrt{\cos (x - \frac{π}{2})} d x

\Rightarrow

\int \sqrt{\sin x} d x = 2 E (\frac{x}{2} - \frac{π}{4} ∣ 2) + C

Commented by MJS_new last updated on 08/Nov/20

(Elliptic Integral)

Commented by peter frank last updated on 08/Nov/20

\int \sqrt{\sin x} d x = ?

Commented by peter frank last updated on 08/Nov/20

thank you