Solve Int. 1: Find x^7, x^5, x^3 Terms in Answer

[elitespart]

1. \int(x^{2} + 5)^{3}dx

This is what the book gives as the answer
1/7x^{7} + 3x^{5} + 25x^{3} + 125x + C

I got something way different. Where are they getting the 3x^5 and 25x^3 from? Thanks.

-v.b.

 

[Mothrog]

What was your result?

 

[elitespart]

Oh, I got: 1/7x^7 + 125x but I distributed the power and started off with x^6 + 125 before anti deriving.

 

[Mothrog]

That would be the problem. You didn't distribute the power correctly. What you did was

(x^2 + 5)^3 = (x^2)^3 + 5^3​

But that's not true. The correct way to expand the power is

(x^2 + 5)(x^2 + 5)(x^2 + 5)​

So, for example

(x+1)^2 = (x+1)(x+1) = x*x + 1*x + 1*x + 1*1 = x^2 + 2x + 1​

Not

(x+1)^2 = x^2 + 1^2​