FIND EACH INDEFINITE INTERGRAL

(X^2+1)^9 X DX

To find the indefinite integral of (x^2+1)^9 dx, we can use the power rule for integration. However, given the complexity of the expression, this integration will require multiple applications of the rule.

1. Start by expanding the expression (x^2+1)^9 using the binomial theorem or Pascal's triangle. The expanded form will be a sum of terms of the form C(n,k) * (x^2)^n * (1^k), where C(n,k) represents the binomial coefficient.

2. Once you have expanded the expression, each term will be of the form C(n,k) * x^(2n) * 1^k. Integrate each term separately using the power rule. Remember that since we are looking for the indefinite integral, we add a "+C" at the end of the solution for each term.

For example, let's consider a simplified case of (x^2+1)^3 x dx:
- The expanded form of (x^2+1)^3 is x^6+3x^4+3x^2+1.
- Now integrate each term separately:
- ∫(x^6) dx = (1/7) * x^7 + C1
- ∫(3x^4) dx = (3/5) * x^5 + C2
- ∫(3x^2) dx = x^3 + C3
- ∫(1) dx = x + C4

To get the solution for the original question (x^2+1)^9 x dx, you will need to go through this process of expanding the expression and integrating each term separately for each term of the expansion. It can be a laborious task due to the high power, but it is not conceptually difficult.