Question

integrate cos squareroot x

Answer 1

2 root x sin rootx plus 2 cos rootx plus c

Answer 2

$\huge{\boxed{\underline{\blue{Solution}}}}$

$∫ {cos}^{2} \sqrt{x} dx \\ using \: identity \: \: cos2x = 2 {cos}^{2} x - 1 \\ then \\ ∫ \frac{cos2 \sqrt{x} + 1 }{2} dx \\ = ∫ \frac{cos2 \sqrt{x} }{2} dx + ∫ \frac{1}{2} dx \\ \\ \frac{1}{2} ∫ \cos(2 \sqrt{x} ) dx \\ \\ take \: 2 \sqrt{x} = t \\ on \: squaring \\ 4x = {t}^{2} \\ 4dx = 2tdt \\ dx = \frac{t}{2} dt \\ now \: using \: above \: derived \: concept \\ = \frac{1}{2} ∫ \cos(t) \frac{t}{2} dt \\ = \frac{1}{4} ∫t \cos(t) dt \\ by \: using \: by \: parts \: we \: get \\ \\ \frac{1}{4} t \sin(t) + \cos(t) \\ now \: ∫ \frac{1}{2} dx = \frac{x}{2} \\ final \: answer - \\ \frac{1}{4} (t \sin(t) + \cos(t) ) + \frac{x}{2} \\ put \: value \: of \: t = 2 \sqrt{x} \\ then \\ \frac{1}{4} (2 \sqrt{x} \sin(2 \sqrt{x} ) + \cos(2 \sqrt{x} ) ) + \frac{x}{2}$


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