USE INTEGRATION BY PARTS TO FIND EACH INTEGRAL
ƪx^5 Lnx dx
Go on :
wolframalpha dot com
When page be open in rectangle type your function.
After few seconds you will see all about your function.
Then click option:
Indefinite integral:Show steps
If you click on left down rectangle of this option you can save solution as image.
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.