How do you find the antiderivative of int (6x)/(x^2-7)^(1/9)dx6x(x27)19dx from [3,4][3,4]?

1 Answer
Feb 1, 2017

The integral is approximately equal to 17.54617.546.

Explanation:

The first thing to do with definite integrals is to make sure that they're in fact definite and not improper.

The integral (6x)/(x^2 -7)^(1/9)6x(x27)19 is continuous on [3,4][3,4]. We are dealing with a definite integral.

I think we should consider a u-substitution to integrate. Let u = x^2 - 7u=x27. Then du = 2xdxdu=2xdx and dx = (du)/(2x)dx=du2x. Furthermore, the new bounds of integration become 22 to 99 because we will now be working in uu.

=>int_2^9 (6x)/u^(1/9) * (du)/(2x)926xu19du2x

=>int_2^9 3/u^(1/9)923u19

=>3int_2^9 u^(-1/9)392u19

=>3[9/8u^(8/9)]_2^93[98u89]92

=>3[9/8(9)^(8/9) - 9/8(2)^(8/9)]3[98(9)8998(2)89]

~~17.54617.546

Hopefully this helps!