Find the indefinite integral and check the result by differentiation. ∫u²√(u^3+2) du
Solve by substitution. Let v = u^3 + 3, and that makes 3 u^2 du = dv. The integral becomes
(1/3)�ç v^1/2 dv
Take it from there.
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To find the indefinite integral of ∫u²√(u^3+2) du, we can use the method of substitution. Here's how we can solve it step by step:
Step 1: Let's start by making a substitution.
Let u^3 + 2 = t, then differentiate both sides with respect to u to find du.
Differentiating both sides gives:
3u² du = dt
Therefore, du = dt / (3u²)
Step 2: Substituting the value of du and u^3 + 2 in terms of t in the original integral:
∫u²√(u^3+2) du = ∫u²√t (dt / 3u²)
Simplifying, we get:
(1/3) ∫√t dt
Step 3: Now, let's integrate (1/3) ∫√t dt with respect to t.
Using the power rule for integration, the integral of t^(1/2) is:
(2/3) t^(3/2) + C
Therefore, (1/3) ∫√t dt = (2/3) t^(3/2) + C
Step 4: Substitute back the value of t in terms of u.
Recall that we made the substitution u^3 + 2 = t.
So, replacing t, we get:
(2/3) (u^3 + 2)^(3/2) + C, where C is the constant of integration.
This is the indefinite integral of ∫u²√(u^3+2) du.
Now, let's verify the result by differentiation:
Step 5: Differentiate the obtained expression with respect to u.
Taking the derivative:
d/dx [(2/3) (u^3 + 2)^(3/2)] = (2/3) * 3/2 * (u^3 + 2)^(1/2) * 3u²
Simplifying, we get:
2(u^3 + 2)^(1/2) * u²
This is the derivative of the expression we obtained from the indefinite integral.
Therefore, we have successfully checked the result by differentiation.