how do you integrate
sqrt( x^2 - (x^2)(e^(-2x)) )?
To integrate the given expression, sqrt(x^2 - (x^2)(e^(-2x))), you can use the technique of u-substitution. Here's how you can proceed:
Step 1: Identify the appropriate u-substitution.
Let's take u = x^2 - (x^2)(e^(-2x)). By differentiating both sides, you can find du.
du = (2x - 2x(e^(-2x)) - 2(x^2)(e^(-2x))) dx.
Step 2: Rewrite the integral in terms of u and du.
Substitute u and du back into the expression to get:
∫sqrt(u) du.
Step 3: Evaluate the u-substituted integral.
To integrate ∫sqrt(u) du, you can use the power rule for integration. The power rule states that for any real number n (except -1), the integral of u^n du is (u^(n+1))/(n+1) + C. Using this rule, you can integrate the expression to get:
(2/3)u^(3/2) + C.
Step 4: Substitute back for u to obtain the final result.
Substitute x^2 - (x^2)(e^(-2x)) back in for u to get the final result:
(2/3)(x^2 - (x^2)(e^(-2x)))^(3/2) + C.
Therefore, the integral of sqrt(x^2 - (x^2)(e^(-2x))) is (2/3)(x^2 - (x^2)(e^(-2x)))^(3/2) + C, where C is the constant of integration.