Please solve it step by step. Integrate:
∫(x^2/(√(x^2+1))dx
To integrate ∫(x^2/(√(x^2+1))dx, we can use a substitution method. Let's introduce the substitution:
Let u = x^2 + 1
Differentiating both sides with respect to x gives:
du/dx = 2x
Solving for dx, we have:
dx = du / (2x)
Now, we can rewrite the integral in terms of u:
∫(x^2/(√(x^2+1))dx = ∫((x^2/(√u))(du/2x)) = (1/2) * ∫((x/(√u)))du
Canceling out the x terms, we get:
(1/2) * ∫(du/√u) = (1/2) * ∫(u^(-1/2))du
Using the power rule for integration, we have:
(1/2) * (2u^(1/2)) + C = √u + C
Finally, substituting back u = x^2 + 1:
∫(x^2/(√(x^2+1))dx = √(x^2 + 1) + C , where C is the constant of integration.