How to integrate cos(x)/x?

To integrate cos(x)/x, use the method of integration by parts.

Integration by parts involves breaking down the integrand into two parts and integrating one part while differentiating the other. The formula for integration by parts is:

∫u dv = uv - ∫v du

where u and v are functions of x.

Let u = cos(x) and dv = 1/x dx. Then, du = -sin(x) dx and v = ln|x|.

Using the formula for integration by parts, we have:

∫cos(x)/x dx = ∫u dv
= uv - ∫v du
= cos(x) ln|x| + ∫sin(x) ln|x| dx

To integrate ∫sin(x) ln|x| dx, use integration by substitution.

Let u = ln|x| and dv = sin(x) dx. Then, du = 1/x dx and v = -cos(x).

Using the formula for integration by parts again, we have:

∫sin(x) ln|x| dx = ∫u dv
= uv - ∫v du
= -cos(x) ln|x| - ∫(-cos(x)/x) dx

To integrate ∫(-cos(x)/x) dx, use integration by parts once more.

Let u = 1/x and dv = -cos(x) dx. Then, du = -1/x^2 dx and v = sin(x).

Using the formula for integration by parts again, we have:

∫(-cos(x)/x) dx = ∫u dv
= uv - ∫v du
= -sin(x)/x - ∫(sin(x)/x^2) dx

To integrate ∫(sin(x)/x^2) dx, use integration by substitution.

Let u = 1/x^2 and dv = sin(x) dx. Then, du = -2/x^3 dx and v = -cos(x).

Using the formula for integration by parts one last time, we have:

∫(sin(x)/x^2) dx = ∫u dv
= uv - ∫v du
= -cos(x)/x^2 + 2∫(cos(x)/x^3) dx

To integrate ∫(cos(x)/x^3) dx, use integration by substitution.

Let u = 1/x^3 and dv = cos(x) dx. Then