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

Answer 1

#= 1/4 x sin 2x + 1/8 cos 2x + x^2/4+ C#

#int dx qquad x * cos^2 (x)#
it will be easier first to use the double angle formula #cos 2A = 2 cos^2 A - 1# or #cos^2 A = (cos 2A + 1)/2#

so we are looking at

#1/2 int dx qquad color{red}{x cos 2x} + x#

as it's a composite term, we should do the red bit using IBP ie

#int u v' = uv - int u' v#

here

#u = x, u' = 1# #v' = cos 2x, v = 1/2 sin 2x#

so

#1/2 int dx qquad color{red}{x cos 2x} + x#
#= 1/2 { 1/2 x sin 2x - int dx qquad 1/2 sin 2x +int dx qquad x }#
#= 1/2 { 1/2 x sin 2x + 1/4 cos 2x + x^2/2 } + C#
#= 1/4 x sin 2x + 1/8 cos 2x + x^2/4+ C#
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Answer 2

The integral of (x \cdot \cos^2(x)) with respect to (x) is:

[ \int x \cdot \cos^2(x) , dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C ]

Where (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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