Maths: Integration

How do I find the integral of 18x(3x+1)^7? I want to use the formula which states that the integral of (ax+b)^n = 1/a x 1/n+1 x (ax+b)^n+1 but that 18x is confusing me, Please help.

(edited 11 months ago)

Reply 1

ghostwalker

Original post by Risermax

How do I find the integral of 18x(3x+1)^7? I want to use the formula which states that the integral of (ax+b)^n = 1/a x 1/n+1 x (ax+b)^n+1 but that 18x is confusing me, Please help.

That formula is the infamous "inverse chain rule", and, no, it won't work here.

Either expand the brackets - rather tedious - or use a substitution; u=3x+1, will vastly simplify things.

(edited 11 months ago)

Reply 2

Risermax

Original post by ghostwalker

That formula is the infamous "inverse chain rule", and, no, it won't work here.

Either expand the brackets - rather tedious - or use a substitution; u=3x+1, will vastly simplify things.

oh ok I figured it out thanks.

Reply 3

davros

Original post by Risermax

How do I find the integral of 18x(3x+1)^7? I want to use the formula which states that the integral of (ax+b)^n = 1/a x 1/n+1 x (ax+b)^n+1 but that 18x is confusing me, Please help.

If you want to avoid substitution and expansion, you could convert your integral into the difference of two integrals where you can apply the rule that you're really desperate to use.

Note that 6(3x+1) = 18x + 6 so you've almost got 6(3x+1)^8 where your rule can be used. So write:

18x(3x+1)^7 = (18x + 6 - 6)(3x+1)^7 = 6(3x+1)^8 - 6(3x+1)^7

and use the infamous "reverse chain rule" on each integral.

Personally I would prefer the sub u = 3x+1 and go from there, but the above approach gives you a second (equivalent) way of checking what's going on :smile:

Reply 4

davros

Original post by Risermax

oh ok I figured it out thanks.

and just to throw in another suggestion - you could do the integral "by parts" by differentiating the 'x' and integrating the (3x+1)^7.

So this is is one of those unusual integrals where you have about 4 different approaches that you can use to cross-check your results :smile: