How do I evaluate #int(13 x^2 + 12 x^-2)dx#?

Question

Emily Ammerman · Accepted Answer

I would apply a sum's integral.

#intf(x)+g(x)dx=intf(x)dx+intg(x)dx#

to split the igral giving argument:

#int13x^2dx+int12x^-2dx=#

and incorporating:

#(13x^(2+1))/(2+1)+(12x^(-2+1))/(-2+1)=13/3x^3-12/x+c#

Luke Alvarez · Answer

undefined To evaluate the integral ∫(13x^2 + 12x^(-2))dx, you can use the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1. Applying this rule to each term separately, we get:

∫(13x^2 + 12x^(-2))dx = (13/3)x^(2+1) + 12(-2+1)x^(-2+1) + C

Simplifying further:

∫(13x^2 + 12x^(-2))dx = (13/3)x^3 - 24x^(-1) + C

Therefore, the integral of (13x^2 + 12x^(-2))dx is equal to (13/3)x^3 - 24x^(-1) + C, where C is the constant of integration.

Isabella Ackerman · Answer

undefined To evaluate the integral ∫(13x^2 + 12x^(-2)) dx, you need to apply the power rule for integration. For each term, add 1 to the exponent and divide the coefficient by the new exponent. So, integrate each term separately:

∫13x^2 dx = (13/3) * x^3 + C
∫12x^(-2) dx = 12 * (x^(-1))/(-1) + C

Combine the results to get the final integral:

∫(13x^2 + 12x^(-2)) dx = (13/3) * x^3 - 12/x + C