integrate: (x^2 + 1)^k dx
To integrate the expression (x^2 + 1)^k with respect to x, we can use the technique of substitution.
Let's denote u = x^2 + 1. Then, du/dx = 2x, which implies du = 2x dx. We can rewrite the given expression in terms of u:
(x^2 + 1)^k dx = (u)^k (du/2)
Now we have transformed our integral in terms of u. To integrate this expression, we can use the power rule for integration. The power rule states that ∫ u^n du = (1/(n+1))u^(n+1) + C, where C is the constant of integration.
Applying the power rule, we get:
∫ (x^2 + 1)^k dx = (1/2) ∫ u^k du
= (1/2) * (1/(k+1)) * u^(k+1) + C
= (1/2(k+1)) * (x^2 + 1)^(k+1) + C
Therefore, the indefinite integral of (x^2 + 1)^k with respect to x is (1/2(k+1)) * (x^2 + 1)^(k+1) + C, where C is the constant of integration.