# How do you find the definite integral #int_(pi/2)^((5pi)/2) x^2 cos (1/5x)dx# ?

Find the indefinite integral

Subtract the indefinite integral evaluated at the lower limit from the indefinite integral evaluated at the upper limit.

Find the indefinite integral:

Use integration by parts:

Substitute into the formula:

Use integration by parts, again:

Substitute into the formula:

Substitute equation [2] into equation [1]:

The last integral is trivial:

Please observe that I have not added a constant of integration, because we are going to use the results to evaluate a definite integral:

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To find the definite integral (\int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} x^2 \cos\left(\frac{1}{5}x\right)dx), you can use integration by parts. Set (u = x^2) and (dv = \cos\left(\frac{1}{5}x\right)dx). Then, differentiate (u) and integrate (dv) to find (du) and (v). After that, apply the integration by parts formula:

[\int u , dv = uv - \int v , du]

Substitute the values and evaluate the integral at the given limits to find the definite integral.

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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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