Find the following indefinite integral: $\int (x^2+6x+5)^{12} (x+3) \ dx$

Question

The solution I got was $(1/13)(x^2+6x+5)^{13} + C$ I am not sure if I am correct though and would like help. Thanks!

Answer 1

Hint: $$\frac{d}{dx}\bigg[\big(x^2 + 6x + 5\big)^k\bigg] = \frac{d}{dx}\bigg[\big((x+3)^2 - 4\big)^k\bigg] = 2k\cdot\big((x+3)^2 - 4\big)^{k-1}\cdot (x+3)$$

Answer 2

Hint:

$$\int (x^2 + 6x + 5)^{12}(x+3) \, \mathrm dx = \frac 1 2 \int (x^2 + 6x + 5)^{12}(2x+6) \, \mathrm dx$$