# How do you integrate #x * cos^2 (x)#?

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To integrate x * cos^2(x), you can use integration by parts. Let u = x and dv = cos^2(x) dx. Then differentiate u to get du/dx = 1, and integrate dv to get v = (1/2) * (x + sin(2x)/2). Apply the integration by parts formula:

∫x * cos^2(x) dx = u * v - ∫v * du

Substitute the values:

= x * (1/2) * (x + sin(2x)/2) - ∫(1/2) * (x + sin(2x)/2) * 1 dx

Then, integrate the remaining integral:

= (1/2) * (x^2/2 + x * sin(2x)/2) - (1/4) * (x^2/2 + x * sin(2x)/2 + cos(2x)/4) + C

Simplify to get the final result:

= (x^2/4) + (x * sin(2x)/4) - (cos(2x)/8) + C

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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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