What is the integral of #(cos(x^(1/2)))#?

Answer 1

# 2 (sqrt(x) sin sqrt(x) + cos sqrt(x) + C )#

obvious sub # q = sqrt(x), dq = (dx)/(2 sqrt(x)) \implies dx = 2 q \ dq#
so the integral is #2 \int \ q cos q \ dq#
integration by parts using form #int u v' = uv - int u' v#
here # u = q, u' = 1# and #v' = cos q, v = sin q#

thus, we now have

# 2 (q sin q - int sin q \ dq )#
# = 2 (q sin q + cos q + C )#
# = 2 (sqrt(x) sin sqrt(x) + cos sqrt(x) + C )#
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Answer 2

The integral of ( \cos(x^{1/2}) ) with respect to ( x ) is an elementary integral and can be expressed in terms of Fresnel integrals. The indefinite integral is given by:

[ \int \cos(x^{1/2}) , dx = 2 \sqrt{x} , \text{FresnelC}\left(\sqrt{\frac{2}{\pi}} \sqrt{x}\right) + C ]

where ( \text{FresnelC}(x) ) denotes the Fresnel cosine integral function and ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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