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How to integrate ln(x)^2*cos(x)?

To integrate ln(x)^2*cos(x), use integration by parts with u = ln(x)^2 and dv = cos(x)dx.

First, find the derivative of u:
du/dx = 2ln(x)(1/x) = 2ln(x)/x

Next, find the antiderivative of dv:
∫cos(x)dx = sin(x)

Using the integration by parts formula, we have:
∫ln(x)^2*cos(x)dx = ln(x)^2*sin(x) - ∫2ln(x)/x*sin(x)dx

To evaluate the second integral, use integration by parts again with u = ln(x) and dv = sin(x)dx.

Find the derivative of u:
du/dx = 1/x

Find the antiderivative of dv:
∫sin(x)dx = -cos(x)

Using the integration by parts formula, we have:
∫2ln(x)/x*sin(x)dx = -2ln(x)*cos(x) + 2∫cos(x)/x dx

To evaluate the second integral, use integration by parts again with u = 1/x and dv = cos(x)dx.

Find the derivative of u:
du/dx = -1/x^2

Find the antiderivative of dv:
∫cos(x)dx = sin(x)

Using the integration by parts formula, we have:
∫cos(x)/x dx = ln(x)*sin(x) + ∫ln(x)/x^2*sin(x)dx

Substitute this result back into the previous equation:
∫2ln(x)/x*sin(x)dx = -2ln(x)*cos(x) + 2(ln(x)*sin(x) + ∫ln(x)/x^2*sin(x)dx)

Finally, substitute both results back into the original equation:
∫ln(x)^2*cos(x)dx = ln(x)^2*sin(x) + 2(ln(x)*cos(x) - ln(x)*sin(x) - ∫ln(x)/x^2*sin(x)dx)

This integral cannot be evaluated using elementary functions, so the answer is:
∫ln(x)^2*cos(x)dx = ln(x)^2*sin(x) + 2(ln(x)*cos(x) - ln(x)*sin(x) - ∫ln(x)/x^2*sin(x)dx) + C

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