Find the indefinite integral.
x^2(5 x^3 + 9)^3 dx
I observed that the derivative of the inside of the bracket sits out front as a multiple of that.
If that is the case, I just reverse by "thinking"
if dy/dx = x^2(5x^3 + 9)^3
y = (x^2/(15x^2)) (5x^3 +9)^4 * (1/4)+ c
= (1/60) (5x^3 + 9)^4 + c
Thanks Reiny for all your help, I really appreciate it.
To find the indefinite integral of the given function, x^2(5x^3 + 9)^3, we can use a combination of power rule and chain rule.
Step 1: Expand the given expression.
(5x^3 + 9)^3 = (5x^3 + 9)(5x^3 + 9)(5x^3 + 9)
Simplifying the expression, we get:
(5x^3 + 9)^3 = (25x^6 + 45x^3 + 45x^3 + 81)(5x^3 + 9)
= (25x^6 + 90x^3 + 81)(5x^3 + 9)
= 125x^9 + 450x^6 + 405x^3 + 225x^6 + 810x^3 + 729
Step 2: Integrate each term separately.
∫x^2(5x^3 + 9)^3 dx = ∫(125x^9 + 450x^6 + 405x^3 + 225x^6 + 810x^3 + 729) dx
= 125∫x^9 dx + 450∫x^6 dx + 405∫x^3 dx + 225∫x^6 dx + 810∫x^3 dx + 729∫dx
Step 3: Integrate each term by applying the power rule.
∫x^n dx = (1/(n+1)) * x^(n+1) + C
Using this rule for each term, we get:
125 * (1/10) * x^10 + 450 * (1/7) * x^7 + 405 * (1/4) * x^4 + 225 * (1/7) * x^7 + 810 * (1/4) * x^4 + 729x + C
Simplifying this expression, we get the final integral:
(12.5x^10 + 64.2857x^7 + 101.25x^4 + 327.857x) + 729x + C
Therefore, the indefinite integral of x^2(5x^3 + 9)^3 is (12.5x^10 + 64.2857x^7 + 101.25x^4 + 327.857x + 729x) + C.