Evaluate the integral?

Question

for the integral sign I use f
f x^2(x^3+5)^6

The answer should be: (1/21) * (x^3+5)^7+C

I only get to the following:
x^3/3 * (x^3+5)^7 / 7

(x^3(x^3+5^7)) / 21

How does that x^3 cross out??

Answer 1

Here's what I got....

∫ x² (x³ + 5)^6 dx

Let u = x³ + 5, du = 3x² dx

Since you need 3x² for the "du" part of the substitution, toss in (1/3) · 3 in front of your integral.

∫ (1/3) 3x² (x³ + 5)^6 dx
= (1/3) ∫ u^6 du
=(1/3) (1/7) u^7 + C
=(1/21) (x³ + 5)^7 + C

It looks like you've integrated the x² part, instead of letting it serve as part of the "du."

Answer 2

ok. well, you would have to use substitution for this. if u substitute
u=x^3 so du = 3x^2 dx

f x^2(x^3 + 5)^6 dx
= f [(u+5)^6]/3 du
= 1/3 f (u+5)^6 du
= 1/3 [(u+5)^7]/7
= 1/21 (x^3 + 5)^7 + C

i really dont know what u did. i dont know how u got to that. but i hope u understand what i did. its integration by substitution, i made up a variable (u) and set it equal to x^3 and then i replaced dx by the appropriate expression involving du and integrated with respect to u. then i put the x^3 back in.

let me know if u need more help