USE INTEGRATION BY PARTS TO FIND EACH INTEGRAL

ƪx^5 Lnx dx

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To find the integral of x^5 * ln(x) dx, we can use the technique of integration by parts. The formula for integration by parts is:

∫(u * v) dx = u * ∫v dx - ∫(u' * ∫v dx) dx

Let's assign u and dv to the expressions in our integral:

u = ln(x) (This is the function we differentiate)
dv = x^5 dx (This is the function we integrate)

Next, we differentiate u to obtain du and integrate dv to obtain v. Let's take the derivatives and integrals:

du = (1/x) dx (Differentiating ln(x) with respect to x)
v = (1/6)x^6 (Integrating x^5 with respect to x)

Now, we can use the integration by parts formula:

∫(x^5 * ln(x)) dx = u * ∫v dx - ∫(u' * ∫v dx) dx

∫(x^5 * ln(x)) dx = ln(x) * (1/6)x^6 - ∫((1/x) * (1/6)x^6) dx

Simplifying the expression, we have:

∫(x^5 * ln(x)) dx = (1/6) * ( x^6 * ln(x) - ∫x^5 dx)

Now, let's integrate the remaining term:

∫x^5 dx = (1/6) * x^6

Substituting this back into the equation, we get:

∫(x^5 * ln(x)) dx = (1/6) * (x^6 * ln(x) - (1/6) * x^6) + C

Thus, the integral of x^5 * ln(x) dx is:

(1/6) * (x^6 * ln(x) - (1/6) * x^6) + C

where C is the constant of integration.